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BRADBURY'S 

ELEMENTARY    ALGEBRA 

DESIGNED   FOR  THE 

USE  OF  HIGH  SCHOOLS  AND  ACADEMIES. 


WILLIAM  F.  BRADBUEY,  A.  M., 

[BAD   MASTER  OF   THE  CAMBRIDGE  HIGH   SCHOOL  ;    AUTHOR  OF  A  TREATISE  OK 
TRIGONOMBTRT  AND   SURVEYING,  AND   OF  AN   ELEMENTARY  GEOMETRY. 


BOSTON: 

PUBLISHED  BY  THOMPSOlSr,  BROWN,  &  CO., 

23  Hawley  Street. 


EDUCATION  DEM 

EATON    AND    BRADBURY'S 

USED   WITH   UNEXAMPLED    SUCCESS   IN    THE    BEST    SCHOOLS   AND 
ACADEMIES  OF  THE  COUNTRY.  )    7    V    / 


( 


Bradbury's  Eaton's  Elementary  ARiTHMETia  -^ 
Bradbury's  Eaton's  Practical  Arithmetic         '■ 

EIaton's  Primary  Arithmetic. 
Eaton's  Elements  of  Arithmetic 

Eaton's  Intellectual  AlRithmetic. 
Eaton's  Common  School  Arithmetic. 
Eaton's  High  School  Arithmetic. 


Bradbury's  Elementary  Algebra. 

Bradbury's  Elementary  Geometry. 

Bradbury's  Elementary  Trigonometry. 

Bradbury's  Geometry  and  Trigonometry,  in  one  volume. 

Bradbury's  Elementary  Geometry.     University  Edition. 
Plane,  Solid,  and  Spherical. 

Bradbury's  Trigonometry  and  Surveying. 

Keys  of  Solutions  to  Practical,  Common  School,  and 
High  School  Arithmetics,  to  Elementary  Algebra, 
Geometry  and  Trigonometry,  and  Trigonometry  and 
Surveying,  for  the  use  of  I'eadiers. 


COPYRICIIT,  186S, 

By  WILLIAM   F.    BRADBURY   AND  JAMES  H.   EATON. 

COPTKIOHT,  1877, 

Bt  WlLLUll   F.  BRADBURY  AND  JAMES  B.  SATOII. 


UMVKESITY    PrK-SS  :    JoHN    WlLhON   AKD  SOK, 

Caubkidgk. 


PREFACE. 


It  was  the  intention  of  the  author  of  Eaton's  Arith- 
metics to  add  to  the  series  an  Algebra,  and  he  had  com- 
menced the  preparation  of  such  a  work.  Although  its 
completion  has  devolved  upon  another,  the  author,  as  far 
as  practicable  in  a  work  of  this  character,  has  followed 
the  same  general  plan  that  has  made  the  Arithmetics  so 
popular,  and  spared  no  labor  to  adapt  the  book  to  the 
wants  of  pupils  commencing  this  branch  of  mathematics. 

A  few  problems  have  been  introduced  in  Section  II.,  to 
awaken  the  pupil's  interest  in  Algebraic  operations,  and 
thus  prepare  him  for  the  more  abstract  principles  which 
must  be  mastered  before  the  more  difficult  problems  can  be 
solved.  Special  attention  is  invited  to  the  arrangement 
of  the  equations  in  Elimination  ;  to  the  Second  Method  of 
Completing  the  Square  in  Affected  Quadratics  ;  and  to  the 
number  and  variety  of  the  examples  given  in  the  body 
of  the  work   and  in   the  closing  section. 

The  Theory  of  Equations,  the  Explanation  of  Negative 
Results,  of  Zero  and  Infinity,  and  of  Imaginary  Quantities, 
are  omitted,  as  topics  not  appropriate  to  an  Elementary 
Algebra.     It  may  also  be  better  for  the  younger  pupils  to 

54!M12 


IV  PREFACE. 

pass  over  the  two  tlieorems  in  Art.  74,  until  they  become 
more  fiiiTjilia:/ with  algebraic  reasoning. 

While  the  book  has  not  been  made  simple  by  avoiding 
the  legitimate  use  of  the  negative  sign  before  a  parenthesis 
or  a  fraction,  the  difficulty  which  is  caused  to  beginners 
by  the  introduction  of  negative  indices  in  simple  division 
has  been  obviated  by  deferring  their  introduction  to  the 
section  on  Powers  and  Roots,  where  they  are  fully  ex- 
plained. 

The  utmost  conciseness  consistent  with  perspicuity  has 
been  studied  throughout  the  work.  It  is  hoped  the  book 
will  commend  itself  to  both  teachers  and  pupils. 

W.  F.  B. 
Cambridge,  Mass.,  May  17, 1868. 


CONTENTS 


SECTION    I. 

Page 

Definitions  and  Notation   . -     7 

The  Signs 7  1  Axioms 9 

SECTION    II. 
Algebraic  Opeeations 10 

SECTION    III. 
Definitions  and  Notation  (Continued  from  Section  I.) 14 

SECTION    IV. 
Addition 19 

SECTION    V. 
Subteaction 25 

SECTION    VI. 
Multiplication -       ....    31 

SECTION    VII. 
Division 37 

SECTION    VIII. 
Demonstration  op  Theoeems 43 

SECTION    IX. 
Factoring 47 

SECTION    X. 
Greatest  Common  Divisor 54 

SECTION    XI. 
Least  Common  Multiple 59 


SECTION    XII. 
Fractions 63 


General  Principles 63 

Signs  of  Fractions 64 

To  reduce  a  Fraction  to  its  lowest  terms  65 
To  reduce  Fractions  to  equivalent  Frac- 
tions having  a  Common  Denominator  66 

To  add  Fractions 68 

To  subtract  Fractions 69 

To  reduce  a  Mixed  Quantity  to  an  Im- 
proper Fraction 70 

To  reduce  an  Improper  Fraction  to  an 

Integral  or  Mixed  Quantity       ...  72 


To  multiply  a  Fraction  by  an  Integral 

.Quantity 73 

To  multiply  an  Integral  Quantity  by 

a  Fraction 75 

To  divide  a  Fraction  by  an  Integral 

Quantity 76 

To  divide  an  Integral  Quantity  by  a 

Fraction 77 

To  multiply  a  Fraction  by  a  Fraction  .  .  77 
To  divide  a  Fraction  by  a  Fraction  .    .      79 


SECTION    XIII. 
Equations  of  thk  First  Degeeb  containing  but  one  Unknown  Quantity 
Definitions 82  1  Reduction  of  Equations 


Transposition    . 83  1  Problems 91 

Clearing  of  Fractions 86  i 


VI  CONTENTS, 

SECTION     XIV. 

Equations  of  the  First  Degree  containing  two  Unknown  Quantities    ....  104 

Elimination  by  Substitution  ....     105  I  Elimination  by  Combination  ....  108 

Elimination  by  Comparison    ....    106  |  Problems 112 

SECTION    XV. 

Equations  op  the  First  Degree  containing   more   than  two  Unknown  Quan- 
tities   118 

SECTION    XVI. 

Powers  and  Roots 125 

Involution  of  Fractions 130 


Negative  Indices 125 

Multiplication  and  Division  of  Powers 
of  Monomials 126 

Transferring  Factors  from  Numerator 
to  Denominator,  or  Denominator  to 
Numerator  of  a  Fraction      ....    127 

Involution  of  Monomials 129 


Involution  of  Binomials 131 

Square  Root  of  Numbers 139 

Cube  Root  of  Numbers 142 

Evolution  of  Monomials 147 

Square  Root  of  Polynomials     ....  148 

To  find  any  Root  of  a  Polynomial    .    .  152 


SECTION    XVII. 
Radicals 154 


Definitions 154 

To  reduce  a  Radical  to  its  Simplest  Form    154 
To  reduce  a  Rational  Quantity  to  the 

form  of  a  Radical 157 

To  reduce  Itadicals  having  different  In- 
dices to  equivalent  ones  having  a 
Common  Index 158 


To  add  Radicals 160 

To  subtract  Radicals 161 

To  multiply  Radicals 162 

To  divide  Radicals 1()3 

To  involve  Rjidicals 166 

To  evolve  Radicals 166 

Polynomials  having  Radical  Terms      .  167 


SECTION    XVIII. 

Pure  Equations  which  require  in  their  Reduction  ErrHER  Involution  or  Evo- 
lution       170 

SECTION   XIX. 

Affected  Quadratic  Equations 178 

Completing  the  Square 178  I  Third  Method  of  Completing  the  Square  186 

Second  Method  ofCompleting  the  Square  182  I  Problems 1^ 

SECTION    XX. 
Quadratic  Equations  contadjino  two  Unknown  Quantities 196 

SECTION    XXI. 
Ratio  and  Proportion 207 

SECTION    \ X  1  I  . 
Aritbhrtical  Progression .    216 

SECTION    XXIII. 
GioiKTRiciJ.  pEooaissioN 225 

SECTION    XXIV. 
Misoellaneous  Exakplis 236 

SECTION     XXV. 
Logarithms 053 


ELEMENTARY    ALGEBRA. 


SECTION    I. 

DEFINITIONS. 

1,  Mathematics  is  the  science  of  quantity. 

2.  Quantity  is  that  which  can  be  measured  ;  as  distance, 
time,  weight. 

3*  Arithmetic  is  the  science  of  numbers.  In  Arithmetic 
quantities  are  represented  by  figures. 

4*  Algebra  is  Universal  Arithmetic.  In  Algebra  quan- 
tities are  represented  by  either  letters  or  figures,  and  their 
relations  by  signs.  ^ 

NOTATION. 

5.  Addition  is  denoted  by  the  sign  -\-,  called  plus ;  thus, 
3  +  2,  i.  e   3  plus  2,  signifies  that  2  is  to  be  added  to  3. 

6t  Subtraction  is  denoted  by  the  sign  — ,  called  minus ; 
thus,  Y  —  4,  i.  e.  t  minus  4,  signifies  that  4  is  to  be  sub- 
tracted from  t. 

7.  Multiplication  is  denoted  by  the  sign  X  ;  thus,  6X5 
signifies  that  6  and  5  are  to  be  multiplied  together.  Be- 
tween a  figure  and  a  letter,  or  between  letters,  the  sign  X 
is  generally  omitted  ;  thus,  6a6  is  the  same  as  6  X  «  X  ^• 
Multiplication  is  sometimes  denoted  by  the  period  ;  thus, 
8  .  6  .  4  is  the  same  as  8  X  6  X  4. 


8  ELEMENTARY   ALGEBRA. 

8.  DiyiSioi^  is  denoted  by  the  sign  -^  ;  thus,  9  -i-  3  sig- 
nifies* tliat»9  is  to  be  divided  by  3.  Division  is  also  iridi- 
L'ateil  .by;  the  IVa^ctional  form  ;  thus,  §  is  the  same  as  9  -f-  3. 

9.  Equality  is  denoted  by  the  sign  =;  thus,  $1  =  100 
cents,  signifies  that  1  dollar  is  equal  to  100  cents.  An  ex- 
pression in  which  the  sign  =  occurs  is  called  an  equa- 
tion, and  that  portion  which  precedes  the  sign  =  is  called 
the  first  member,  and  that  which  follows,  the  second  mem- 
ber. 

10.  Inequality  is  denoted  by  the  sign  >  or  <,  the 
smaller  quantity  always  standing  at  the  vertex  ;  thus, 
8  >  6  or  6  <  8  signifies  that  8  is  greater  than  6. 

11.  Three  dots  .'.  are  sometimes  used,  meaning  hence, 
therefore. 

12.  A  Parenthesis  (  ),  or  a  Vinculum •,  indicates 

that  all  the  quantities  included,  or  connected,  are  to  be 
considered  as  a  single  quantity,  or  to  be  subjected  to  the 
same  operation  ;  thus,  (8  +  4)  X  3  =  12  X  3,  or  =  24 
-f  12  =  36  ;  21  —  6  -r-  3  =  15  -^  3,  or  =  T  —  2  =  5.. 
Without  the  parenthesis,  these  examples  would  stand 
thus  :  8  +  4  X  3  =  8  +  12  =  20  ;  21  —  6  ^  3  =  21 
—  2  =  19;  the  sign  X.  in  the  former,  not  affecting  8  ; 
nor  the  sign  -r-,  in  the  latter,  21. 


Examples. 

1.  9  +  7  —  3  +  4  =  how  many? 

2.  (9  +  15)   -4-  3  ==  how  many? 

3.  — - —  X   14  =  how  many  ? 

4.  (14  +  13)   X   (6  —  2j  =  how  many? 

5.  10  +  (7  —  4)  -4-  3  X  4  =  how  many? 

6.  26  —  (6+  7)  =  how  many? 

7.  150  —  (18  —  11)  =  how  many? 


DEFINITIONS.  9 

8.  Prove  that  n5  +  8  —  49  —  14  +  190  —  54  —  16. 

9.  Prove  that  216  —  44  +  14  >    144  +  13  —  15. 

10.  Place  the  proper  sign  (=,  >,  or  <)  between  these 
two  expressions,   (247  -f-  104)  and  (546  —  195). 

11.  Place  the  proper  sign  (=,  >,  or  <)  between  these 
two  expressions,   (119  —  41  +  16)   and  (311  —  104). 

12.  Place  the  proper  sign  (=,  > ,  or  <)  between  these 
two  expressions,  (417  +  31)  —  (187  —  72)  and  (127  + 
179). 

AXIOMS. 

13*  All  operations  in  Algebra  are  based  upon  certain 
self-evident  truths  called  Axioms,  of  which  the  following 
are  the  most  common  :  — 

1.  If  equals  are  added  to  equals  the  sums  are  equal. 

2.  If  equals  are  subtracted  from  equals  the  remainders 
are  equal. 

3.  If  equals  are  multiplied  by  equals  the  products  are 
equal. 

4.  If  equals  are  divided  by  equals  the  quotients  are 
equal. 

5.  Like  powers  and  like  roots  of  equals  are  equal. 

6.  The  whole  of  a  quantity  is  greater  than  any  of  its 
parts. 

7.  The  whole  of  a  quantity  is  equal  to  the  sum  of  all 
its  parts. 

8.  Quantities  respectively  equal  to  the  same  quantity 
are  equal  to  each  other. 


10  ELEMENTARY  ALGEBRA. 


SECTION   II. 

ALGEBRAIC    OPERATIONS. 
II,   A  Theorem  is  something  to  be  proved. 

15.  A  Problem  is  something-  to  be  done. 

16,  The  Solution  oif  a  Problem  in  Algebra  consists,  — 
1st.   In  reducing  the  statement  to  the  form  of  an  equa- 
tion ; 

2d.  In  reducing  the  equation  so  as  to  find 'the  value 
of  the  unknown  quantities. 

Examples  for  Practice. 

1.  The  sum  of  the  ages  of  a  father  and  his  son  is  60 
years,  and  the  age  of  the  father  is  double  that  of  the  son  ; 
what  is  the  age  of  each  ? 

It  is  evident  that  if  we  knew  the  age  of  the  son,  by 
doubling  it  we  should  know  the  age  of  the  father.  Sup- 
pose we  let  X  equal  the  age  of  the  son;  then  2x  equals 
the  age  of  the  father;  and  then,  by  the  conditions  of  the 
problem,  a:,  the  son's  age,  plus  2j7,  the  father's  age,  equals 
60  years;  or  3a:  equals  60,  and  (Axiom  4)  x,  the  son's 
age,  is  ^  of  60,  or  20,  and  2  a;,  the  father's  age,  is  40 
Expressed  algebraically,  the.  process  is  as  follows:  — 

Let        X  =  son's  age, 

then     2x  =  father's  ago. 

X  -\-  2x  =  60, 

Sx  =  60, 

X  =  20,  tlie  son's  age. 

2  J'  =  10,   thp  fiilhfM-'rt  age. 


DEFINITIONS.  11 

2.  A  horse  and  carriage  are  together  worth  $450  ;  but 
the  horse  is  worth  twice  as  much  as  the  carriage  ;  what 
is  each  worth?  Ans.   Carriage,  $160;  horse,  $300. 

All  problems  should  be  verified  to  see  if  the  answers 
obtained  fulfil  the  given  conditions.  In  each  of  the  pre- 
ceding problems  there  are  two  conditions,  or  statements. 
For  example,  in  Prob.  2  it  is  stated  (1st)  that  the  horse 
and  carriage  are  together  worth  $450,  and  (2d)  that  the 
horse  is  worth  twice  as  much  as  the  carriage  ;  both  these 
statements  are  fulfilled  by  the  numbers  150  and  300. 

3.  The  sum  of  two  numbers  is  72,  and  the  greater  is 
seven  times  the  less  ;  what  are  the  numbers  ? 

4.  A  drover  being  asked  how  many  sheep  he  had,  said 
that  if  he  had  ten  times  as  many  more,  he  should  have 
440  ;  how  many  had  he  ? 

5.  A  father  and  son  have  property  of  the  value  of 
$8015,  and  the  father's  share  is  four  times  the  son's; 
what  is  the  share  of  each? 

Ans.   Father's,   $6412;   son's,   $1603. 

6.  A  farmer  has  a  horse,  a  cow,  and  a  sheep  ;  the  horse 
is  worth  twice  as  much  as  the  cow,  and  the  cow  twice 
as  much  as  the  sheep,  and  all  together  are  worth  $490  ; 
how  much  is  each  worth  ? 

OPERATION. 

Let        X  ■=  the  price  of  the  sheep, 

then    2j;  =    *'       "       "     "  cow, 

and      4a?  t=    "       "       "     "  horse; 
and  their  sum  7  x  :^  490, 

•       j7  =  70,  the  price  of  the  sheep, 

and  2ip  =  140,  "       "       "  "    cow, 

and  4^  =  280,  *'       "       "  "     horse. 


12  ELEMENTARY  ALGEBRA. 

7.  A  man  has  three  horses  which  are  together  worth 
$540,  and  their  values  are  as  the  numbers  1,  2,  and  3; 
what  are  the  respective  values  ? 

Let  X,  2  X,  and  3  x  represent  the  respective  values. 

Ans.  $90,  $180,  and  $270. 

8.  A  man  has  three  pastures,  containing  360  sheep, 
and  the  numbers  in  each  are  as  the  numbers  1,  3,  and  5  ; 
liow  many  are  there  in  each  ? 

9.  Divide  63  into*  three  parts,  in  the  proportion  of  2, 
3,  and  4. 

Let  2x,  3x,  and  4x  represent  the  parts. 

10.  A  man  sold  an  equal  number  of  oxen,  cows,  and 
sheep  for  $1500;  for  an  ox  he  received  twice  as  much 
as  for  a  cow,  and  for  a  cow  eight  times  as  much  as  for 
a  sheep,  and  for  each  sheep  $  6  ;  how  many  of  each  did 
he  sell,  and  what  did  he  receive  for  all  the  oxen  ? 

Ans.   10  of  each,  and  for  the  oxen,  $960. 

11.  Three  orchards  bore  872  bushels  of  apples;  the  first 
bore  three  times  as  many  as  the  second,  and  the  third 
bore  as  many  as  the  other  two  ;  how  man}'  bushels  did 
each  bear  ? 

12.  A  boy  spent  $4  in  oranges,  pears,  and  apples;  he 
bought  twice  as  many  pears  and  five  times  as  many  apples 
as  oranges ;  he  paid  4  cents  for  each  pear,  3  for  each 
orange,  and  1  for  each  apple  ;  how  many  of  each  did  he 
buy,  and  how  much  did  he  spend  for  oranges  ?  how  much 
for  pears,  and  how  much  for  apples  ? 

.        (25  oranges,  50  pears,  and  125  apples. 

(  Spent  for  oranges,  $0.75  ;  pears,  $2  ;  apples,  $1.25. 

13.  A  farmer  hired  a  man  and  two  boys  to  do  a  piece 
of  work  ;  to  the  man  he  paid  $12,  to  one^boy  $6,  and 
to  the  other  $  4  per  week ;  they  all  worked  the  same 
time,  and  received  $264;  how  many  weeks  did  they 
work  ?  Ans.   12  weeks. 


DEFINITIONS.  10 

14.  Three  men,  A,  B,  and  C,  agreed  to  build  a  piece 
of  wall  for  $99;  A  could  build  1  rods,  and  B  6,  while 
C  could  build  5  ;  how  much  should  each  receive  ? 

15.  Four  boj^s,  A,  B,  C,  and  D,  in  counting  their  money, 
found  they  had  together  $1.98,  and  that  B  had  twice  as 
much  as  A,  C  as  much  as  A  and  B,  and  D  as  much  as 
B  and  C  ;  how  much  had  each  ? 

Ans.   A   18  cents,   B  36,  C  54,  and  D  90. 

16.  It  is  required  to  divide  a  quantity,  represented  by 
a,  into  two  parts,  one  of  which  is  double  the  other. 

OPERATION. 

Let        X  ==:  one  part, 
then    2x  =  the  other  part. 

Sx  =  a, 

X  =  -,  one  part, 

o 

2x  =  -— ,  the  other  part. 

o 

17.  If  in  the  preceding  example  a  =  24,  what  are  the 
required  parts  ? 

A  «  24  .  .2a  48 

^"^-  3  ==  T  =  ^'  ^^^    T  =  Y=^^- 

18.  It  is  required  to  divide  c  into  three  parts  so  that 

the  first  shall  be  one  half  of  the  second  and  one  fifth  of 

the  third.  ,  c      2c  .5c 

Ans.    -,    -,    and    -. 

19.  Divide  n  into  three  parts,  so  that  the  first  part 
shall  be  one  third  the  second  and  one  seventh  of  the 
third. 

20.  A  is  one  half  as  old   as  B,  and  B    is  one  third  as 

old  as   C,    and  the  sum  of  their   ages    is  p  ;  what   is    the 

age  of  each  ?  ^^^    ^,^  p     g,^  ^_   ^^^  ^'s  ^^ 

y  9  y 


14  ELEMENTARY   ALGEBRA. 


SECTION    III. 

DEFINITIONS    AND    NOTATION. 

[Continued  from  Section  I  ] 

17.  The  last  letters  of  the  alphabet,  ar,  y,  z,  &c.,  are 
used  in  algebraic  processes  to  represent  unknown  quanti- 
ties, and  the  first  letters,  a,  b,  c,  &c.,  are  often  used  to 
represent  known  quantities. 

Numerical  Quantities  are  those  expressed  by  figures,  as 
4,  6,  9. 

Literal  Quantities  are  those  expressed  by  letters,  as 
a,  X,  y. 

Mixed  Quantities  are  those  expressed  by  both  figures 
and  letters,  as  3  a,  4  a:. 

18.  The  sign  plus,  -\-,  is  called  the  positive  or  affirm- 
ative sign,  and  the  quantity  before  which  it  stands  a  pos- 
itive or  affirmative  quantity.  If  no  sign  stands  before  a 
quantity,  -\-  is  always  understood. 

19.  The  sign  minus,  — ,  is  called  the  negative  sign,  and 
the  quantity  before  which   it  stands,  a  negative  quantity. 

20.  Sometimes  both  +  and  —  are  prefixed  to  a  quan- 
tity, and  the  sign  and  quantity  are  both  said  to  be  am- 
biyuoxis;  thus,  8  zb  3  =:  II  or  6,  and  a  zh  6  =  a  -j-  6, 
or  a  —  h,  according  to  circumstances. 

21.  The  words  plus  and  minus,  positive  and  negative, 
and  the  signs  -|-  and  — ,  have  a  merely  relative  signifi- 
cation ;  thus,  the  navigator  and  the  surveyor  always  rep- 
resent their  northward  and  eastward  progress  by  the  sign 
-|-,  and  their  southward  and  westward  progress  by  the 
sign  — ,  though,  in  the  nature  of  things,  there  is  nothing 
to  prevent  representing  northings  and  eastings  by  — , 
and  southings  and  westings  by  -("•     So  if  a  man's  prop- 


DEFINITIONS.  15 

erfy  is  considered  positive,  his  gains  should  also  be  con- 
sidered positive,  while  his  debts  arid  his  losses  should  be 
considered  negative ;  thus,  suppose  that  I  have  a  farm 
worth  $5000  and  other  property  worth  $3000  and  that 
I  owe  $1000,  then  the  net  value  of  my  estate  is  $5000 
+  $3000  —  $1000  —  $7000.  Again,  suppose  my  farm 
is  worth  $5000  and  my  other  property  $3000,  while 
I  owe  $12000,  then  my  net  estate  is  worth  $5000 
_|_  $3000  —  $12000    =  —  $4000,    i.    e.    I    am    worth 

—  $4000,  or,  in  other  words,  I  owe  $4000  more  than  I 
can  pay.     From  this  last  illustration  we  see  that  the  sign 

—  may  be  placed  before  a  quantity  standing  alone,  and 
it  then  merely  signifies  that  the  quantity  is  negative, 
without  determining  what  it  is  to  be  subtracted  from. 

22t  The  Terms  of  an  algebraic  expression  are  the  quan- 
tities which  are  separated  from  each  other  by  the  signs 
+  or  —  ;  thus,  in  the  equation  4a  —  b  z^  Sx  -\-  c  —  1  y, 
the  first  member  consists  of  the  two  terms  4  a  and  —  b, 
and  the   second  of  the  three  terms  3  x,  c,  and  —  7  y. 

23.  A  Coefficient  is  a  number  or  letter  prefixed  to  a 
quantity  to  show  how  many  times  that  quantity  is  to  be 
taken;  thus,  in  the  expression  4:X,  which  equals  x  -\-  x 
-^  X  c\-  X,  the  4  is  the  coefficient  of  x ',  so  in  3  a  b,  which 
equals  ab  -\-  ab  ~\-  ab,  3  is  the  coefficient  of  a6;  in4a&, 
4  a  may  be  considered  the  coefficient  of  b,  or  4  6  the  co- 
efficient of  a,  or  a  the  coefficient  of  4  6. 

Coefficients  may  be  numerical  or  literal  or  mixed ;  thus, 
in  4 a 6,  4  is  the  numerical  coefficient  of  ab,  a  is  the  lit- 
eral coefficient  of  4  6,  4  a  is  the  mixed  coefficient  of  6. 

If  no  numerical  coefficient  is  expressed,  a  unit  is  un- 
derstood ;  thus,  X  is  the  same  as  Ix,  be  as  16c. 

24.  An  Index  or  Exponent  is  a  number  or  letter  placed 
after  and  a  little  above  a  quantity  to  show  how  many  times 
that  quantity  is  to  be  taken  as  a  factor;  thus,  in  the  ex- 


16  '  FXEMENTARY   ALGEBRA. 

pression  W,  which  equals  6  X  ^  X  &,  the  3  ia  the  index  or 
exponent  of  the  power  to  which  b  is  to  be  raised,  and  it 
indicates  that  h  is  to  be  used  as  a  factor  3  times. 

An  exponent,  like  a  coefficient,  may  be  numerical,  lit- 
eral, or  mixed  ;  thus,  ^^  a:",  ar*",  &c. 

If  no  exponent  is  written,  a  unit  is  understood  ;  thus 
6  =r  6\  a  =  a},  &c. 

Coefficients  and  Exponents  must  be  carefully  distin- 
guished from  each  other.  A  Coefficient  shows  the  num- 
ber of  times  a  quantity  is  taken  to  make  up  a  given 
sum  ;  an  Exponent  shows  how  many  times  a  quantity  is 
taken  as  a  factor  to  make  up  a  given  product  ;  thus 
i:X=:^x-\-x-[-x-\-x,  and  x*^=xy^xy^xy^x. 

25*  The  product  obtained  by  taking  a  quantity  as  a 
factor  a  given  number  of  times  is  called  a  power,  and 
the  exponent  shows  the  number  of  times  the  quantity  is 
taken. 

26.  A  Root  of  any  quantity  is  a  quantity  which,  taken 
as  a  factor  a  given  number  of  times,  will  produce  the 
given  quantity. 

A  Root  is  indicated  by  the  radical  sign,  v',  or  by  a 
fractional  exponent.  When  the  radical  sign,  \/,  is  used, 
the  index  of  the  root  is  written  at  the  top  of  the  sign, 
though  the  index  denoting  the  second  or  square  root  is 
generally  omitted  ;  thus, 

a/ X,  or  xh,  means  the  second  root  of  x ; 
^Ic,  or  xi,       "         "    third         "      "    a:,  &c. 

Every  quantit}'-  is  considered  to  be  both  the  first  power 
and  the  first  root  of  itself 

27.  The  Reciprocai.  of  a  quantity  is  a  unit  divided  by 
that  quantity.     Thus,  the  reciprocal  of  6  is  -,  and  of  x,  -. 

0  X 


DEFINITIONS.  17 

2S»  A  Monomial  is  a  single  term ;  as  a,  or  3  x,  or 
bhxy. 

29.  A  Polynomial  is  a  number  of  terms  connected 
with  each  other  by  the  signs  plus  or  minus  ;  slq  x  -\-  y, 
or  3a  -\-  4:X  —  1  ahy. 

30.  A  Binomial  is  a  polynomial  of  two  terms ;  as 
^x  -\-  ^y,  ov  X  —  y. 

31.  A  Residual  is  a  binomial  in  which  the  two  terms 
are  connected  by  the  minus  sign,  as  x  —  y. 

32.  Similar  Terms  are  those  which  have  the  name  powers 
of  the  same  letters,  as  x  and  3  a;,  or  bai?  and  —  2ax^. 
But  X  and  x\  or  5  a  and  5  b,  are  dissimilar. 

33.  The  Degree  of  a  term  is  denoted  by  the  sum  of 
the  exponents  of  all  the  literal  factors.  Thus,  2  a  is  of 
the  first  degree  ;  3  a^  and  A:  ah  are  of  the  second  de- 
gree ;  and  6  a^  x^  is  of  the  seventh  degree. 

34.  Homogeneous  Terms  are  those  of  the  same  degree. 
Thus,  A:a^x,  Sabc,  x^y,  are  homogeneous  with  each  other. 

35.  To  find  the  numerical  value  of  an  algebraic  expres- 
ision  when  the  literal  quantities  are  known,  we  must  sub- 
stitute the  given  values  for  the  letters,  and  perform  the 
operations  indicated  by  the  signs. 

The  numerical  value  of  t  a  —  5*  -|-  c^  when  a  =  4, 
ft  =  2,  and  c  =  5  is  7  X  4  —  2*  +  5^  =  28  —  16 
+  25  =  37. 

Examples. 
Find  the  numerical  values  of  the  following  expressions, 
when  a  =  2,  J  =  13,  c  r::^  4,  cZ  =  15,  m  =  5,  and  n  =:  7. 

1.  a  -\-  h  —  c  -\-  2d.  Ans.  41. 

2.  a^  -f  36c  —  2cd.  Ans.  40. 

3.  1^1  +  ^'  Ans.  219. 


18  ELEMENTARY  ALGEBBA. 


4. 
6. 
6. 

1. 

(a2  _  c  +  h)  (m  +  w). 
^__^    X   (^  -  m  +  .0. 

Ans.  86t. 

8. 
9. 

3aV6  —  c  X  4nv^25m.  ' 

Ans.  1. 

5m—  6n-h3d 
10. -. s- .  Ans.  4. 


11.  v^  —  ^  +  \/Tn. 

12.  (6  —  a)  {d  —  c)  —  m.  Ans.  116. 

13.  13  (4<Z)  4-  4<?  —  ta. 


14.  4a6  +  \/100c  —  -^rf  —  n.  Ans.  122. 

15.  4  a  V60l  +  5a2  62. 

16.  h  —  a  —  {d  —  n).  Ans.  3. 
n.  6  —  a  —  d  —  n.  Ans.  —  11. 

18.  (6  —  o)   (^  —  n).  Ans.  88. 

19.  {h  —  a)  d  —  n.  Ans.  158. 


20.   a  +  6>v/10  (d  —  m)  +  14\/c. 
36*    Write  in  algebraic  form  :  — 

1.  The    sum    of  a    and    6    minus   the   difference    of  m 
and  n.     (m  >  w.) 

2.  F'our  times  the  square  root  of  the  sum  of  a,  b,  and  c. 

3.  Six  times*  the  product  of  the  sum  and  difference  of 
c  and  rf.     (c  >  rf.) 

4.  Five  times  the  cube  root  of  the  sum  of  a,  m,  and  n. 

5.  The  sum  of  m  and  n  divided  by  their  difference. 

6.  The  fourth  power  of  the  difference  between  a  and  m. 


ADDITION. 


19 


SECTION  IV. 

ADDITION. 

37.  Addition  in   Algebra  is   the  process  of  finding  the 
aggregate  or  sum  of  several  quantities. 

For  convenience,   the  subject  is  presented  under  three 
cases. 

CASE  I.      . 

38.  When  the  terms  are  similar  and  have  like  signs. 
1.  Charles  has  6  apples,    James  4  apples,  and    William 

6  apples  ;  how  many  apples  have  they  all  ? 


OPERATION. 


6  apples, 
4  apples, 


or,  letting  a 


5  apples,  ^    represent 

one  apple, 

15  apples. 


6  a 

4  a 

5  a 

15  a 


It  is  evident  that  just  as 
6  apples  and  4  apples  and 
5  apples  added  together 
make  15  apples,  so  6  a  and 
4  a  and  5  a  added  togeth- 
er make  15  a. 


In    the    same    way   —  6  a    and  —  4  a   and   —  5  a   are 
equal  together  to  —  15  a. 

Therefore,   when   the   terms   are  similar   and   have   like 
signs  : 

RULE. 

Add  the  coefficients,  and  to  their  sum  annex  the  common 
letter  or  letters,  and  prefix  the  common  sign. 


(2.) 

(3.) 

(4.) 

(5.) 

(6.) 

(7.) 

b  ax 

Sa' 

4:X 

6y 

—  ^7? 

-bby 

%ax 

^a\ 

X 

lOy 

—   2x^ 

-2hy 

4:ax 

la' 

hx 

y 

—    7:r^ 

-    hy 

2ax 

Sa' 

Zx 
13  a; 

^y 

—    4^:^ 

-    hy 

19  ax 

—  16x« 

20 


ELEMENTARY   ALGEBRA. 


8.  What  is  the  sum  oi  ax^,  Sax^,  2a2p,  and  4  a  a:*? 

Ans.   10  a  x^. 

9.  What  is  the  sum  of  ^bx,  4tbx,  6bx,  and  bx? 

10.  What  is  the  sum  of  2x2/;  Qxy,  10 a: 5/,  and  Sxy? 

11.  What  is  the  sum   of — Ixz,  — xz,   — 4zxz,    and 

—  xz?  Ans.  —  13x2. 

12.  What    is    the    sum    of   —2  b,    —3  b,    —6  b,    and 

—  Sb? 

13.  What  is  the  sum  of  — a  be,  — Sabcy  — 4:  a  be, 
and  — a  be? 

CASE   II. 

39.  When  the  terms  are  similar  and  have  unlike  signs. 

1.  A  man  earns  7  dollars  one  week,  and  the  next  week 
earns  nothing  and  spends  4  dollars,  and  the  next  week 
earns  6  dollars,  and  the  fourth  week  earns  nothing  and 
spends  3  dollars ;  how  much  money  has  he  left  at  the 
end  of  the  fourth  week  ? 

If  what  he  earns  is  indicated  by  -\-,  then  what  he 
spends  will  be  indicated  by  — ,  and  the  example  will 
appear  as  follows  :  — 


+  7  dollars, 

—  4  dollars, 
+  6  dollars, 

—  3  dollars, 

+  6  dollars, 


OPERATION. 

or,  letting  d 
represent    . 
one  dollar, 


[+1  d 

—  4rf 
+  6d 

—  Sd 

+  Qd 


Earning  7  dollars  and 
then  spending  4  dollars, 
the  man  would  have  3 
dollars  left;  then  earn- 
ing 6  dollai-s,  he  would 
have  9  dollars;  then 
spending  3  dollars,  he 
would  have  left  6  dol- 
lars ;  or  he  earns  in  all  7  dollars  -j-  6  dollars  =13  dollars ;  and  spends 
4  dollars  -|-  3  dollars  =  7  dollars;  and  therefore  has  left  the  differ- 
ence between  13  dollars  and  7  dollars  =  6  dollars;  hence  the  sum 
of -f  7  </,  —  4  rf,  -f  6  rf,  and  —  3  rf  is  -f-  6  r/. 

Therefore,  when  the  terms  are  similar,  and  have  unlike 
signs : 


ADDITION.  21 

RULE. 

Find  the  difference  between  the  sum  of  the  coefficients  of 
the  positive  terms,  and  the  sum  of  the  coefficients  of  the  neg- 
ative terms,  and  to  this  difference  annex  the  common  letter 
or  letters,  and  prefix  the  sign  of  the  greater  sum. 


(2.) 

(3.) 

(4.) 

(5.) 

3xy 

4.f 

ISabc^ 

Ux^y 

xy 

-2y^ 

6abc^ 

.  Ux^y 

—  b  xy 

ly^ 

—     a  be' 

lOx^y 

1  xy 

-32/^ 

—  8  a  6  c^ 

x^y 

-  2xy 

Uy" 

4.abc^ 

24:x'y 

4:xy 

Uabc' 

(6.) 

(^•) 

2bxyz 

8  (x 

+  y) 

-  50  xyz 

—  4.{x 

+  y) 

lOxyz 

1(x 

+  y) 

-  61  xyz 

—  3  (a: 

+  y) 

Sxyz 

-      (^ 

+  y) 

—  *l4iXyz  1  {x -\- y) 

8.  Find  the  sum  of  Sx^y^,  — lAx'^y^,  Vlx^y"^,  and  — x^y"^. 

9.  Find   the    sum    of  1{x-\-y),    %{x-\-y),    — (^  +  y), 
and4(x-}-y)-  Ans.   18(a?+y). 

10.  Find  the  sum  of  —  ax^,  -\- a  oi?,  —  10  a  x^,  -|-  25  a  x^, 
and  —  13  ax}. 

11.  Find  the  sum  of  21  ah,  —  34  a  &,   —  150  ah,  21  a  h, 
and  —  13  ah.  Ans.  —  143  a  h. 

12.  Find    the    s.um    of   ax^,    — 14  a  x^,    11  a  x^,    — ax^, 
44  a  x^,  and  —  a  a:^ 

13.  Find    the    sum    of  11{a  +  h),    —(a-}-h),    (a  +  h), 
and   —  13  (a  4-  h).  Ans.  4  (a  +  h). 


22  ELEMENTARY   ALGEBRA. 

CASE     III. 
40»    To  find  the  sum  of  any  algebraic  quantities. 

The  sum  of  5  a  and  6  6  is  neither  11a,  nor  116,  and 
can  only  be  expressed  in  the  form  of  b  a  -\-  Qb,  or  6  i 
-|-  5  a ;  and  the  surti  of  5  a  and  —  46  is  b  a  —  46;  but  in 
finding  the  sum  of  6  a,  6  6,  5  a,  and  — 4  6,  the  a's  can  be 
added  together  by  Case  I.,  and  the  6's  by  Case  II.,  and 
tlie  two  result^  connected  by  the  proper  sign ;  thus,  5  a 
_[_66-f5a  —  46=:10a  +  2  6. 

1.  Find  the  sum  of  6  rf,  —  2  6,  x,  3y,  ox,  —  3  6,  3  6c 
+  4  c?,  bx,  T  6  +  2  X,  and  —  3  6  c. 

OPERATION.  Poj.    convenience,   simi- 

6rf  —  26-|-     a:  +  3^-|-3  6c  lar  terms   are  written   un- 

4rf_36  +  5a;              — 36c  der  each  other ;    then  by 

_4_  Y  A  _[_  5  -P  Case   I.   the    first    column 

\    n  at  the  left  is  added ;  the 

jL second    by   Case    II.,  and 

10rf+26+13a:  +  3y  soon;  +36c  and  — 36c 

cancel. 

This  cise  includes  the  two  preceding  cases,  and  hence 
to  find  the  sum  of  any  algebraic  quantities : 

RULE. 

Write  similar  terms  under  each  other,  find  the  sum  of 
each  column,  and  connect  the  several  sums  with  their  proper 
signs. 

(2.) 
4a;—  Ta  +  Sy  — 46  +  3r 
6a—     y  -f.  46  — 2r 
4  a  — 2f/  +  86—     z 
—  3a  —86  —  10c 

4x  —10c 


ADDITION.  23 

(3.) 

—  3^+    3  c— 1  s/x^  +  y 
—  Wc  +  Ss/^—y 
d\/x 

7a  +     b—lOc -\-6\/lc 

4.  Add    together   T  V^,    —  8  .t,    1  x'^,    —  6\/^    4ar', 

—  8  a;,  4:X,  and  7  a:^,  —  S  \/  x.  

Ans.   18  2^2  —  12  a:  —  t  \/T. 

6.  Add  together  3aa;  —  4:ab  -{-  2xi/,  1  ab  -{-  bx  —  4«, 
*l xy  —  3  a  +  4ar,  and  -\-  abc  —  ax  -\-  6xy. 

6.  Add  together  1  x  —  Say  —  5  ab  -\-  4:C,  3«a;-|-4a; 
-\-  bab  —  5c,  and  3c  —  Sax  -\-  1  y  -\-  c. 

Ans.   Ua;  —  Say  -\-  Sc  -\-  1  y. 

7.  Add   together   5a  —  32-)-7a7-]-4aa:  —  Sab,    bab 

—  5a  +  22  —  4aa:  +  4,    and    6  —  2  a6  +  3:tr  +  4y + 
4aar. 

8.  Add  together  Qxy  -\-  Qxz  —  6m?z-f-4«,  i^mn  — 
Sxy  -\-2n  —  8m  w,  —  Qxz  -\-  4:n  —  Sxy  -\-  6,  and  10  mn 

—  lOn  +  3  —  9.  Ans.  0. 

9.  Add  together  8aw+19war—  55  6  +  c,  —  19v  + 
146 — 16c-j-y,  and  ISnx  —  44am-|-15u  —  4y. 

10.  Add  together  17  aar^  +  19aar^  —  14ax*  +  16  aa:^, 
1 3  a  a:''   —  5  a  x^   -j-  6  a  a:^  —  10  a  ar',   and    14:  ax*  -j-  17  a  a:^ 

—  3aar'+  15aa;2.  Ans.   71  a^r^  +  19  0^^-^  —  5a:c*. 

11.  Add  together  m -{- n  —  4a  +  6c  —  7y,  8c  —  4?w 
-|-3n  —  5a  -|-  3c,  7a  —  17c  +  1  y  —  10m  —  6w,  and 
14 n  — 8a  —  7c4-  102^  —  8  m. 

12.  Add  together  Saxy-{-11bxy —  16ca:y  —  9axy, 
IQbxy —  18ca?3^  -f-  lOaajy —  14:axz,    IQcxy  -\-  2baxy 

—  ^bxy  -\-  2bcxy,    and     lOaxz  +  Shxy  ; —  lOcx^  -f- 
Aaxz.  Ans.   Siiaxy -{- 29bxy  —  Sexy. 


24  ELEMENTARY   ALGEBRA. 

13.  Add  together  3  (x  +  y),  —  4  (a:  +  y),  and  1  (x  -\-  y). 

Ad8.  Q  {x  -\-  y). 

14.  Add    together   6(2a;+y  — 3z),   and  —2{2x-\-y 

—  3z). 

Note.  —  If  several  terms  have  a  common  letter  or  letters,  the 
sura  of  their  coefficients  may  be  placed  in  parenthesis,  and  the  com- 
mon letter  or  letters  annexed ;  thus, 

6x-f8x  — 5a:=(6-f-8  — 5)a:; 
ax  -^^bx  —  2cx^=(a-\-Zh  —  2c)x; 
he  xy  -\-  adxy  —  ac  xy  =  (b  c  -\-  ad  —  a  c)  xy. 

15.  Add  together  ax —  hx  -^  ^x,  and  2ax  -\-  i:bx  —  x. 

Ans.   (3a  +  36  +  2)  X. 

16.  Add  together  by  — 3cy  -\-bay,  and  cy  +4Ay 
"2  cry. 

17.  Add  together  2xy  —  axy,  and  ^xy  —  Saxy. 

Ans.   (8  —  4cr)  xy. 

18.  Add  together  T(3x  +  5y)  +  3a  — 6a:  +  8a6,  Sx 
-(-6(3a:   +  5y)    +1a   —bab,    and   8a;   +2(3a:  +  5y) 

—  7  a  —  3  a  6.  Ans.  47  a:  -(-  70y  +  3  a. 

41.  From  what  has  gone  before,  it  will  be  seen  that 
addition  in  Algebra  differs  from  addition  in  Arithmetic. 
In  Arithmetic  the  quantities  to  be  added  are  always  con- 
sidered positive  ;  while  in  Algebra  both  positive  and  neg- 
ative quantities  are  introduced.  In  Arithmetic  addition 
always  implies  augmentation  ;  while  in  Algebra  the  sum 
may  be  numerically  less  than  any  of  the  quantities  added  ; 
thus,  the  sum  of  10 a:  and  — 8a:  is  2x,  which  is  the 
numerical  difference  of  the  two  quantities. 


SUBTRACTION.  25 

SECTION   V. 

SUBTRACTION. 

42.  Subtraction  in  Algebra  is  finding"  the  difference 
oetween  two  quantities. 

1.  John  has  6  apples  and  James  has  2  apples ;  how 
niany  more  has  John  than  James  ? 

Let  a  represent  one  apple,  and  we  have 

6a] 

2aj.,  or  6a  —  2a  =  4a. 

4aJ 

2.  During  a  certain  day  A  made  9  dollars  and  B  lost 
6  dollars  ;  what  was  the  difference  in  the  profits  of  A 
and  B  for  the  day  ?  If  gain  is  considered  -|-,  then  loss 
must  be  considered  — ,  and  letting  d  represent  one  dol- 
lar, it  is  required  to  take  —  6  d  from  9  d. 


OPERATION. 

9d 
—  6d 


It  is  evident  that  the   difference  be- 
tween   A's  and  B's   profits  for  the  day 
13   9d  -^  Gd  =  15d;    that    is,    dd  — 
15  rf  (— 6J)  =  9rf-f  6(/=  15c?. 

Hence  it  appears  that,  as  addition  does  not  always  im- 
ply augmentation,  so  subtraction  does  not  always  imply 
diminution. 

Subtracting  a  positive  quantity  is  equivalent  to  adding  an 
equal  negative  quantity;  and  subtracting  a  negative  quan- 
tity is  equivalent  to  adding  an  equal  positive  quantity. 

Suppose  I  am  worth  $1000;  it  matters  not  whether  a 
thief  steals  $400  from  me,  or  a  rogue  having  the  author- 
ity involves  me  in  debt  $400  for  a  worthless  article  ;  for 


26  ELEMENTARY   ALGEBRA. 

in  either  case  I  shall  be  worth  only  $600.  The  thief  sub- 
tracts  a  poailwe  quantity  ;  the  rogue  adds  a  negative  quan- 
tity. 

Again,  suppose  I  have  $1000  in  my  possession,  but 
owe  $400;  it  is  immaterial  to  me  whether  a  friend  pays 
the  debt  of  $400  or  gives  me  $400  ;  for  in  either  case  I 
shall  be  worth  $1000.  In  the  former  case  the  friend 
subtracts  a  vegaiive  quantity ;  in  the  latter,  he  adds  a  pos- 
itive.    Or,  to  make  the  proof  general : 

1st.    Suppose  -|-  h  to  be  taken  from  a  -{-  b 

the  result  will  be  a ; 

and  adding  —  b  to  a  -\-  b  we  have  a  -{-  b  —  b, 

which  is,   as  before,  equal  to  a. 

2d.   Suppose  —  ft  to  be  taken  from  a  —  h 

the  result  will  be  a  ; 

and  adding  -\-  b  to  a  —  b  we  have  a  —  b  -^  b, 

which  is,  as  before,  equal  to  a, 

3.  Subtract  b  -\-  c  from  a. 

OPERATION.  b  subtracted   from  a  gives 

a  —  (^b  -\-  c)  =■  a  —  b  —  c  a  —  ft;   but  in   subtracting  ft 

we  have  subtracted  too  small 
a  quantity  by  r,  and  therefore  the  remainder  is  too  great  by  c,  and 
the  remainder  sought  is  a  —  ft  —  c. 

4.  Subtract  ft  —  c  from  a. 

OPERATION.  In  subtracting  ft  from  a  we 

a  —  (ft  —  c)  =  a  —  b  -\-  c         subtract  a  quantity  too  great 

by  c ;  therefore  the  remainder 
(a  —  ft)  would  be  just  so  much  too  small,  and  the  remainder  sought 
is  a  —  h  -{-  c. 

43*  By  examining  the  examples  just  given  it  will  be 
seen  that  in  every  case  the  sign  of  each  term  of  the 
subtrahend  is  changed,  and  that  the  subsequent  process 
is  precisely  the  same  as  in  addition  ;  hence,  for  subtrac- 
tion in  Algebra  we  have  the  following 


SUBTRACTION.  27 

RULE. 

Change  the  sign  of  each  term  of  the  subtrahend  from  -f- 
to  — ,  or  —  to  -\-,  or  suppose  each  to  be  changed,  and  then 
proceed  as  in  addition. 


(1.) 

(2.) 

(3.) 

(4.) 

(5.) 

(6.) 

0') 

Min. 

9 

9 

9 

9 

9 

9 

9 

Sub. 

9 

6 

3 

0 

—-3 

—  6 

—  9 

Rem.      0  3  6  9  12  15  18 

In  examples  1-7,  the  minuend  remaining  the  same  while 
the  subtrahend  becomes  in  each  3  less,  the  remainder  in 
each  is  3  greater  than  in  the  preceding. 


(8.) 

(9.) 

(10.) 

(11.) 

(12.) 

(13.) 

(14.) 

Min.      9 

6 

3 

0      • 

—    3 

—    6 

—    9 

Sub.      9 

9 

9 

9 

9 

9 

9 

Rem.     0      —  3      —  6      —  9      —  12      —  15      —  18 

In  examples  8-14,  the  minuend  in  each  becoming  3  less 
while  the  subtrahend  remains  the  same,  the  remainder  in 
each  is  3  less  than  in  the  preceding. 

(15.)    (16.)    (11.)    (18.)        (19.)         (20.)         (21.) 
Min.      9  6  3  0  — 3  — 6  — 9 

Sub.       9  6  3  0  — 3  — 6  — 9 

Rem.     0  0  0  0  0  0  0 

In  examples  15-21,  both  minuend  and  subtrahend  de- 
creasing by  3,  the  remainder  remains  the  same. 

(1.)  (2.)  (3.)  (4.)  (5.)  (6.) 

Min.    26x  21  axy     —Uab     —18c  4=9x1/     —4386 

Sub.     lOx     —   4axy  4a3     —    6c     — 25a;^  21b 

Rem.    I6x  Slaxy     —-llab     — 12c 


28  ELEMENTARY    ALGEBRA. 


a.)      (8.) 

(9.)        (10.)        (11.)        (12.) 

Min.         lOx    —  ^axy 

4ai    __    6c    — 25a;y           27* 

Sub.         26a;         21axy    - 

-\Zah    —18c         49a:y    —438* 

Rem.  —16a;    —Zlaxy 

17a*         12c 

(13.) 

(14.) 

Min.  6a;—  14^  +  32 

7«+  18*—  10c 

Sub.    3a;+    3y+     z 

—  25*-f    6c— 8rf 

Rem.  3a;—  ny-f-2z 

7a  +  43*— 16c4-8rf 

15.    From  28  a;  take  — 

17  X.                            Ans.  45  X. 

16.    From  —  347  a  take  223  «.                 Ans.  —  6t0  a. 

17.    From  —  76^/  take 

—  33y.                    Ans.  —  43y. 

18.    From  44  6  take  — 

150*.                          Ans    194*. 

19.  From  —417  c  take  984  c. 

20.  From  —  84z  take  —117  2;. 

21.  From  17  aa;  —  18  *c  +  44a:y  take  25*c  — 14a;y 
+  20  a.  Ans.   17  aa;  — 43*c  +  58a;y  —  20  a. 

22.  From  384a;—  74y  +  18  c  take  118  a;  +  74y  —  27  c. 

Ans.  266  a;  —  148  y  +  45  c. 

23.  From  a:^  —  y^  +  a;^  —  lOa^  take  a;^ -f  4 y' —  a;* — 
43;^.  Ans.  2a;*  —  6  a:^  —  6/. 

24.  From  6a*y  —  4a;y-|-3a;2  take  — 4a  *y  —  Zxz 
—  4a;y. 

25.  From  a;''  +  2  a; y  +  y^  take  x^ —  2xy  ■\' y'^. 

26.  From  a;«  +  2  a;y  +  y«  take  —  x"  -^^2  xy  —  y: 

44*  The  subtraction  of  a  polynomial  may  be  indicated 
by  enclosing  the  polynomial  in  a  parenthesis  and  prefix- 
ing the  sign  — . 

Thus,  ar*  -j-  t/  —  :^  taken  from  7?  —  a^  may  be  written 
a^'  — 2«— (ar'  +  y  — 2'''). 


SUBTRACTION.  29 

When  a  parenthesis  with  the  sign  minus  before  it  is  re- 
moved, the  sign  of  each  term  within  the  parenthesis  must  be 
changed  according  to  the  Bute  for  subtraction. 

Thus,    x'--z'—{:>^  +  f  —  z')  =  x'  —  ^  —  2^—f  + 

And  conversely, 

A  polynomial,  or  any  number  of  the  terms  of  a  polyno- 
mial, can  be  enclosed  in  a  parenthesis  and  the  minus  sign 
placed  before  the  parenthesis  without  changing  the  value  of 
the  expression,  providing  the  signs  of  all  the  terms  are 
changed  from  plus  to  minus  or  from  minus  to  plus. 

Thus,  a'  —  b'  +  c'  -\-  d  —  X  =  a^  —  (b'  ~  c'  -  d  +  x). 

Note.  —  When  the  sign  of  the  first  term  in  the  parenthesis  is 
plus,  the  sign  need  not  be  written.     (Art.  18.) 

According  to  this  principle  a  polynomial  can  be  writ- 
ten in  a  variety  of  ways. 

Thus,  a^—3x^y+  ^^y'—i  =  o(^  —  {Zx'^y —  ^ xy""  +  f) 

r=^-^x''y-{-^xy^  +  f) 
=  a?  +  ^xf—{^x^y+f) 
=zx^—f—{3x''y  —  Sxy'')&c. 

Remove  the  parenthesis,  and  reduce  each  of  the  follow- 
ing examples  to  its  simplest  form. 

1.  a^  —  (2ab-\-  c^).  Ans.   a'^—2ab  —  c\ 

2.  x'^  —  6ax-^a^  —  6x^y  —  {x'^-{-  6  ax -\- x'^ —  6  x'^ ij). 

Ans.  —  12  ax. 

3.  rn^  —  n''-\-2x—(4:m'^  +  Sn^  —  4.c). 

4.  IQxy  +  Uc —  ISy  —  {^Uc -]-21  y —16xy). 

Ans.   S2xy -\-2Sc —  4:5y. 

6.  4:X^y—(Sxy^~1x''y'^-\-Sx^y). 
b.  — (-0:^+7  -25xy  +  /). 


ZO  ELEMENTARY    AL(]KBRA. 

Place  in  parenthesis,  with  the  nigii  —  prefixed,  without 
changing"  the  value  of  the  expression, 

1.  The  last  three  terms  of  1  x^  —  14  xy  —  3  2?  -[-  4y. 

Ans.   *ra:2— (14a:y  +  32  — 4y). 

2.  The  last  three  terms  of  ar^  -|-  3/^  —  Sxy  -\-  4:C. 

Ans.  x'^  —  (3  ary  —  y^  —  4  ^). 

3.  The  last  four  terms  of  4:  a  —  1  b  —  6  c  —  S  d  -\-  x\ 

4.  The  last  four  terms  of  a^  -{-  b^ -\- c'' —  d^  +  ct\ 

5.  Write  in  as  many  forms  as  possible  by  enclosing 
two    or   more    of  the    terms   in    parenthesis,    a^  —  b^  -\-  c^ 

45*  In  subtraction,  when  two  quantities  have  a  com- 
mon factor  their  diflference  is  the  difference  of  the  coef- 
ficients of  the  common  factor  multiplied  by  this  factor. 

Thus,  ax  —  bx  =  (a  —  b)  x. 

1.  From  a  x^  take  c  x'^  —  dx^.       Ans.   (a  —  c  -\-  d)  x^. 

2.  From  4  \/ a:  take  ats/ x  •\- b \f  x. 

Ans.   (4  —  a  —  &)  \/  x. 

3.  From  a  a^  take  bx^  —  b  x^.     Ans.  {a  —  b)x^-\-b  x\ 

4.  From  4 a:^  —  6x  take  ax^  -\-  bx. 

Ans.    (4  —  a)x^—{6Jf-  b)  x. 

6.  From  6  a*  +  4  a^  —  a  take  a^  x  —  a^y  -\-  az. 

Ans.    (6  —  x)  a«  +  (4  +y)  a^  —  (1  +  t)  n. 

6.  From  ab  —  be  take  ^b  -\-  ex. 

*l.  From    a^  —  bx  -{-  c >^  x  take  bx'^  -\-  ex  —  rf\/x. 

8.  From  xy"^  +  a:^  —  x'^y'^  take  f  +  x' y  —  x'^y\ 


MULTIPLICATION.  31 

SECTION    VI. 

MULTIPLICATION. 

46.  MuLTTPLicATiox  is  a  short  method  of  finding  the 
sum  of  the  repetitions  of  a  quantit}?-. 

47.  The  multiplier  must  always  be  an  abstract  num- 
ber, and  the  product  is  always  of  the  same  nature  as  the 
multiplicand. 

The  cost  of  4  pounds  of  sugar  at  17  cents  a  pound  is 
17  cents  taken,  not  4  pounds  times,  but  4  times;  and 
the  product  is  of  the  same  denomination  as  the  multi- 
plicand 17,  viz.  cents. 

In  Algebra  the  sign  of  the  multiplier  shows  whether 
the  repetitions  are  to  be  added  or  subtracted. 

L  (+a)X(+4)  =  +  4a; 

i.   e.  -j"  ^  added  4  times  is  -\-a-\-a-\-a-{-a^=-{-4ia. 

2.  (+a)X(~4):::.:  — 4a; 
i.  e.  +  a  subtracted  4  times  is  —  a  —  a  —  a  —  a  =  —  4  a. 

3.  (— «)  X  (4-*)  =  — 4a; 
i.  e.   —  a  added  4  times  is  —  a  —  a  —  a  —  a  =  —  4  a. 

4.  (— «)X(— 4)==  +  4a; 
i.  e.  —  a  subtracted  4  times  is-f-<^  +  <^+«  +  «  =  +  4a. 

In  the  first  and  second  examples  the  nature  of  the 
product  is  -f-  ;  in  the  first,  the  -f-  sign  of  4  shows  that 
the  product  is  to  be  added,  and  +  4a  added  is  -|-  4a; 
in  the  second,  the  —  sign  of  4  shows  that  the  product 
is  to  be  subtracted,  and  -|-  4  a  subtracted  is  —  4  a.  In 
the  third  and  fourth  examples  the  nature  of  the  product 
is  — ;  in  the  third,  the  -\-  sign  of  4  shows  that  the  prod- 
uct  is   to   be  added,  and  —  '4  a   added   is  —  4  a ;  in   the 


82  ELEMENTARY  ALGEBRA. 

fourth,  the  —  sig-n  of  4  shows  Ihat  the  product  is  to  be 
subtracted,  and  —  4  a  subtracted  is  -j-  4  a. 

48t  Ilence  in  multiplication  we  have  for  the  sign  of 
the  product  the  following 

RULE. 

Lake  signs  give  +  ;  unlike,  — . 

Hence  the  products  of  an  even  number  of  negative  fac- 
tors is  positive,  of  an  odd  number,  negative. 

49.  Multiplication  in  Algebra  can  be  presented  best 
under  three  cases. 

CASE    I. 
50*   When  both  factors  are  monomials. 

1.  Multiply  3  a  by  2  b. 

OPERATION. 

3aX25  =  3X^X2x*=3X2XaXJ  =  6c6. 

As  the  product  is  the  same  in  whatever  order  the  factors  Are 
arranged,  we  have  simply  changed  their  order  and  united  in  one 
product  the  numerical  coefficients. 

Hence,  when  both  factors  are  monomials, 

RULE. 
Annex  the  product  of  the  literal  factors  to   the  prodwf 
of  their  coefficients,   remembering   that    like   signs    give  -(' 
and  unlike,  — . 

2.  Multiply  n^  by  a\ 

OPERATION. 

a^  X  "'  =  («  X  «  X  a)  X  (a  X  «)  =  a  X  «  X  n  X  «  X  a  =  a* 
As  the  exponent  of  a  quantity  shows  how  many  times  it  is  taken 
as  a^factor,  a"  =  a  X  «  X  a ;  and  c^  =  a  X  <i'i  and  a'  X  a'  =  «  X 
a  X  a  X  a  X  rt»  and  this  is  equal  to  a*.  (Art.  24.)     Hence, 

Powers  of  the  same  quantity  are  multiplied  together  by 
adding  their  exponents. 


MULTIPLICA'J 

riON. 

33 

(3.) 

(4.) 

(5.) 

(6.) 

a.) 

4,xy 

bx:'f 

"lab 

14  m  n^ 

—     aH' 

Sab 

Ibx^f 

—    SaH 

6  aw* 

—  4:aH 

Vlahxy 

—  66  a«  52 



S4:amnJ 

4.aH' 

8.  Multiply  x"^  by  a?. 

9.  Multiply  x"^  by  x\ 

10.  Multiply  a^  by  —  a, 

11.  Multiply  —  a*  by  a®. 

12.  Multiply  —  c8  by  —  c*. 

13.  Multiply  ^xy'^hy  lax. 

14.  Multiply  504  a^  5^  by  —  8  a'  J. 

15.  Multiply  — 2bxyz  by  ^xyz. 

16.  Multiply  —  417  a  i  c2  by  —  3  a  &2  ^. 

17.  Multiply  together  444  a;  y,  Sx^y^,  and  — 2  2. 

Ans.  —2664  0^3^2;. 

18.  Multiply  4.an^cd  hy  —^^abc'iP. 

19.  Multiply  -— 5  x"*  by  —  6  x".  Ans.  30  a?'"+". 

20.  Multiply  together  14  a  i  c-,  —  5  d^  b  c,  and  —  4  a  hK 

21.  Multiply  2bs/ay  by  Sbx,  Ans.  1bbx\/ay. 

22.  Multiply  4  (x  +  y)  by  3  (x  +  2^). 

Ans.   12  {x  +  y)2. 

Note.  —  Any  number  of  terms  enclosed  in   a  parenthesis  may 
be  treated  as  a  monomial. 

23.  Multiply  —  12  (a2  —  V")  by  —  4  {a"  —  P). 

Ans.  48  (a"  —  I^)\ 

24.  Multiply  (a  —  x)*  by  (a  —  a;)^. 

25.  Multiply  4  (a  +  b)"'  by  2  (a  +  by. 

Ans.  8  (a  +  5)'"+^ 

26.  Multiply  a«  (a:  +  zf  by  a  i^  (a:  +  ^). 

2*  c 


34 


ELEMENTARY    ALGEBRA. 


CASE    II. 
51*    When  only  one  factor  is  a  monomial. 


1.   Multiply  8  +  5  by  3. 

OPERATION. 


8  + 

51 

8  + 

5 

8  + 

5 

24  + 

16 

+    5  =13 


or 


24  +  15  =  39 


2.   Multiply  8  —  0  by  3. 


In  this  example,  not  the 
sum  of  3  repetitions  of  8 
only^  but  of  8  and  5,  is  re- 
quired ;  the  sum  of  3  repeti- 
tions of  8  =  24  ;.  of  3  repeti- 
tions of  5  =  15.  Hence,  the 
sum  of  3  repetitions  of  8  -f- 
5  =  24  -|-  15. 


OPERATION. 


8  — 

5] 

8  — 

5 

8  — 

5 

24  — 

15. 

8  — 


I  or 


24  —  15  =  9 


The  sum  of  3  repetitions 
of  8  =  24 ;  but  it  is  not  the 
sum  of  3  repetitions  of  8  that 
is  required,  but  of  a  num- 
ber 5  units  less  than  8 ;  24, 
therefore,  will  have  in  it  the 
sum  of  5  units  repeated  3 
times,  or  15,  too  much;  the  product  required,  therefore,  is  24  —  15. 

Therefore, 

The  product  of  the  sum  is  equal  to  the  sum  of  the  prod- 
ucts, and  the  product  of  the  difference  to  the  difference  of 
the  products. 


3.    Multiply  X  -{-  1/  —  2  by  a. 

OPERATION. 


ax  +  oy  —  az 


The  sum  of  the  repetitions 
of  a:  a  times,  of  y  a  times,  and 
of  —  z  a  times  is  a  x  -j-  a  f/ 
—  az. 


RULE. 
Multiply  each  term  of  the  multiplicand  by  the  multiplier, 
and  connect  t}ie  several  results  by  their  proper  signs. 


MLLTIPLICATION.  35 

(4.)  (5.) 

Sax  —  4:X 


12  a  x^  —  24:  a  x^ -{-  4:2  a  X  ^  —  4:a^x-{-8adx  —  4:lr^x. 

6.  Multiply  5  m  w  +  4  m^  —  6  n^  by  4  a  6. 

7.  Multiply  16  a^  X  —  Sxz-{-4.^  by  — 3a: y. 

8.  Multiply  ia^  —  cx'^-\-dx  by  — a^. 

9.  Multiply  —QSxy—Ux  —  Qz  by  —  4 2:. 

Ans.  252xyz  +  6Qxz  -\-24: z\ 

10.  Multiply  14  a*  —  13  a'^  +  12  a-^  —  11  a  by  4  a'. 

11.  Multiply  a:  —  2  a  +  14  by  ax. 

12.  Multiply  11  ax  —  I4:by  -\-  llcz  by  —  4.abcxyz. 

13.  Multiply  2I«n2  — 3x2^2_4  5^  by  —^axy. 

CASE    III. 
52t    When  both  factors  are  polynomials. 

1.  Multiply  7  +  4  by  5  —  3. 

OPERATION.  Multiplying   7  -|-  4  by  5  is 

•7    I    4  =11  taking  the  multiplicand  3  too 

c  q  0  many    times ;    therefore,    the 

'. true  product  will  be  found  by 

35  +  20  —  21  —  12       =22  subtracting     3  (7 -j- 4)    from 

5(7  +  4). 

2.  Multiply  X  —  y  hj  a  -\-  h. 

OPERATION.  a  times  x  —  y  =  ax  —  ay\  but 

^ y  X  —  y  is  to  be  taken,  not  a  times 

^    I    ^  only,   but   a  -\- b  times;    therefore, 

a(x  —  y)  is  too  small  hy  h  (x  —  y); 

ax  —  a y  -\-  b X  —  by  g^j^^j  ^j^g  product  required  is  a r  — 

ay  -\-bx  —  by.     Hence, 

RULE. 

Multiply  each  term  of  the  multiplicand  hy   each  term   of 
the  multiplier,  and  find  the  sum  of  the  several  products. 


id  ELLMENTAEY  ALGEBRA. 

(3.)  (4.) 

a^  -\-  2  a  X  -\~  xi^  xy  -\-  ab 

a  —  X  xy  —  ab 


c^  -\-2a^x  -\-     ax^  x^y"^  -\-  ahxy 

—    ci^  X  —  2  a  a;^  —  a:''  —  ahxy  —  a^  b^ 

6.  Multiply  a2  +  J2  _  ^2  by  a^  +  c\ 

6.  Multiply  x^  —  2xy  -\-  y^  by  x^  -\-  y^. 

Ans.  x*  —  2x^y-\-2x"y''  —  2xy'^  +  y*, 

1.  Multiply  4:a*  — 2  aH  +  SaH^  hy  2a^  —  2b\ 

Ans.  8a«  — 4a^6  — 2an2-j-4a»*»— 6a2J4. 

8.  Multiply  ar*  +  2:r^  +  3ar2+2ar+l  by  ar^  — 2x+l. 

9.  Multiply  x^ -\- y^ -\-  z'^  —  xy  —  xz  —  yzhy  x  -\-y  -\-  z. 

Ans.  x^  -\-  ^  -\-  2^  —  3  xy.z. 

10.  Multiply  4:X^—1x^+l0x^—Ux^  by  3  ar  —  2. 

11.  Multiply  a^  +  a2  —  1  by  a^  —  1. 

12.  Multiply  x^+1  ax  —Ua^  by  x  —  *l  a. 

Ans.  x^  _  49  a^  a:  —  14  a^  X  +  98  a\ 

13.  Multiply  X  -\-  y  —  a  by  x  —  ^  +  «. 

14.  Multiply  a"  +  i"  by  a"*  —  b"". 

Ans.  a  »'  +  «  -f  a"*  5"  —  a"  6'"  —  i'"  +  *. 

15.  Multiply  7ar;y  —  14  a-^y^  _|_  21  a:*/  by  Q  x y  —  S. 

Ans.  _21a:y +  84a:2/—  UTar^/H-  126ar*y. 

16.  Multiply  6aH  —  0aP—l2aH^  by  2nb  —  S  H'. 
Ans.   12  a^  62  _  3Q  ^2  ^3  _  24  ^3  ^  _|_  27  a  ^4  _^  3g  ^a  ^4^ 

11.  Multiply  x*  —  x'^  +  x-  —  x+l  by  ar  +  1. 

Ans.  a^  +  h 

18.  Multiply  x'^x'^  +  x'^  —  x+l  by  ar  —  1. 

Ans.  r'  —  2  ar*  +  2  r^  —  2  ar2  +  2  a:  —  1. 

19.  Multiply  X*  +  x'  +  X-  +  X+  1  by  x  +  1. 

20.  Multiply  a:<  +  a:"  +  a:^  4-  x  +  1  by  ar  —  1. 


DIVISION.  37 


SECTION   VIT. 

DIVISION. 

5B.  Division  is  finding  a  quotient  which,  multiplied  by 
the  divisor,  will  produce  the  dividend. 

In  accordance  with  this  definition  and  the  Rule  in 
Art.  48,  the  sign  of  the  quotient  must  be  +  when  the 
divisor  and  the  dividend  have  like  signs ;  —  when  the 
divisor  and  the  dividend  have  unlike  signs  ;  i.  e.  in  di- 
vision as  in  multiplication  we  have  for  the  signs  the  fol- 
lowing 

RULE. 

Like  signs  give  -\- ;  unlike,  — . 

CASE    I. 
54.    When  the  divisor  and  dividend  are  both  monomials. 

1.    Divide  6ab  by  2b. 

OPERATION.  The  coefficient  of  the  quotient  must 

6a6-r-2^=3a  be  a  number  which,  multiplied  by  2, 

the  coefficient  of  the  divisor,  will  give 

6,  the  coefficient  of  the  dividend ;  I.  e.  3 :  and  the  literal  part  of 

the  quotient  must  be  a  quantity  which,  multiplied  by  b,  will  give  a  b ; 

i.  e.  a :  the  quotient  required,  therefore.  Is  3  a. 

Hence,  for  division  of  monomials, 

RULE. 

Annex  the  quotient  of  the  literal  quantities  to  the  quotient 
of  their  coefficients,  remembering  that  like  signs  give  -f-  and 
unlike,  — . 


38  ELEMENTARY   ALGEBRA- 

2.   Divide  a*  by  c?, 

OPERATION. 

a^  -^  a^^=c^  For  a'  x  a^  =  a^    (Art.  50.)     Hence, 

Powers  of  the  same  quantity  are  divided  by  each  other  by 
subtracting  the  exponent  of  the  divisor  from  that  of  the 
dividend. 

(3.)  (4.) 

^Ix^f  4Sa^xy 

• =.Zxy  =:  —  3  ay 

^  xy  —  16  ax 

(5.)  (6.) 

—  276ar^y  ^IQa^x'^yz 

=  —  6a?  =4:ax 


4:6  x^y  —  4:axyz 

1.  Divide  Siaby^  by  2  ay. 

8.  Divide  291  x^  f  by  —99xy^.  Ans.  —Sxy. 

9.  Divide  —  14:xy^  z  hy  2  x  y. 

10.  Divide  —lUa^b^x  by  —2iabx.  Ans.  6ab. 

11.  Divide  a  a:*  by  ax^, 

12.  Divide  8a:«  by  —8  a:*. 

13.  Divide  —  210  x'' y  by  42a:«y. 

14.  Divide  —270  a  5  a:  by  —  135a&ar. 

15.  Divide  —  474a3^^c2  by  158  a  ^c. 

16.  Divide  a;"*  by  x*.  Ans.  a?*""". 

17.  Divide  14a'"a;"  by  — 7a'*a:.      Ans.  — 2a'"-''a:"-^ 

18.  Divide  —l^laH'c'd'  by  S^aHcd^ 

19.  Divide  12  (a:  +  .y)^  by  4:{x  +  y).     Ans.  3(a:+y). 

20.  Divide  —21  (a  — by  by  9{a  —  b)\ 

21.  Divide  {b  —  c)'  by  (^  —  c)^  Ans.   (6  —  c)^ 

22.  Divide  14  (x  —  yY  by  7  (a;  —  yY. 


DIVISION.  39 

CASE    II. 
55*    When  the  divisor  only  is  a  monomial. 

I.  Divide  ax-\-ai/-\-az  hy  a. 

OPERATION.  jjj  t}jg  multiplication  of  a  poly- 

a)  a  X  ~\-  a  y  -\-  a  z  nomial  by  a  monomial,  each  terra 

X  -\-      y  -\-     z  °^  ^^^   multiphcand  is   multiplied 

by  the  multiplier ;   and  therefore 

we  divide  each  term  of  the  dividend  ax  -\-  ay  -\-  az  by  the  divisor 

a,  and  connect  the  partial  quotients  by  their  proper  signs.     Hence, 

RULE. 
Divide  each  term  of  the  dividend  by  the  divisor,  and  con- 
nect the  several  results  by  their  proper  signs. 

(2.)  (3.) 

S  a)  6  a x^  — 24.  ax'  _-  5  x'^y)  —  15  a^ i/  —  2b  x^ y 

2^2  —  8ar^  Bx  +  5 

4.  Divide   l2aQ[^  —  24  a  a?^  -f-  42  a  x  ^  by  Sax. 
6.  Divide  —  4:a^x-\-Sabx—r4:b'^x  by  —  4:X. 

6.  Divide  Q  a^x*  —  12  a!' x^  +  lb  a^  x^  hy  3  a^  ^^.a. 

Ans.  2  x^  —  4  «  X  +  5  a^  a^. 

7.  Divide     12  a*/  —  16  a^/  +  20a«/  —  28  a^    ^J 
4  a*/. 

8.  Divide  —  5  x^  +  10  x^  —  15  x  by  —  5  x. 

9.  Divide  273  (a  +  xf  —  91  (a  +  x)  by  91  (a  +  x). 

Ans.   3(a  +  a;)  —  l  =  3a  +  3x—  1. 
10.  Divide  20  a6c  —  4ac  -(-  8  ac<Z  —  12  a^c^  by —4 ac. 

II.  Divide  16  a^a;^— 32  a'x^  +  48  a*x*  by  lea^a;^ 

12.  Divide  72  x^  /  —  36  x^  /  —  54  x^  /  2^  by  —  18  x^  /. 

Ans.  —4:  +  2xy+Sxz\ 

13.  Divide  IS  a  x^ y -^  54:  x^  y""  +  lOS  c  x^  if  by  9x'^y. 

14.  Divide  40  aH^  +  S  a^  b^  —  96  a«  b^  x^  by  8  a^  i^. 

15.  Divide  39  x*  z^  —  65  a  x«  2^^  +  13  x»  z^  ^y  —  13  x^  z^ 


40  ELEMENTARY   ALGEBRA. 

CASE     III. 

56.  When  the  divisor  and  dividend  are  both  polyno- 
mials. 

1.   Divide  a?  —  Zx^ y  ^Zxy^  —  if  by  x^  —  2xy-\-y^. 

OPERATION. 

3?^1xy-\-y'^)x'  —  Zx''y-\-Zxy^^f  (x  —  y 
a?^2x'^y-\-     xf- 

—     x^y  -\-1xy'^  —  f 

The  divisor  and  dividend  are  arranged  in  the  order  of  the  powers 
of  a:,  beginning  with  the  highest  power,  a:',  the  highest  power  of  x 
in  the  dividend,  must  be  the  product  of  the  highest  power  of  x  in 

the  quotient  and  a:*  in  the  divisor ;  therefore,  —=.  x  must  be  the 

highest  power  of  x  in  the  quotient.  The  divisor  x'  —  2  x  y  -|~  y'  "^"1" 
tiplied  by  x  must  give  several  of  the  partial  products  which  would 
be  produced  were  the  divisor  multiplied  by  the  whole  quotient. 
When  {j?  —  2xy  -{-if)  x  =  x^  —  1^-  y  -\-  xrf  is  subtracted  from 
the  dividend,  the  remainder  must  be  the  product  of  the  divisor  and 
the  remaining  terms  of  the  quotient ;  therefore  we  treat  the  remain- 
der as  a  new  dividend,  and  so  continue  until  the  dividend  is  ex- 
hausted. 

Hence,  for  the  division  of  polynomials  we  have  the  fol- 
lowing 

RULE. 

Arrange  the  divisor  and  dividend  in  the  order  of  the 
powers  of  one  of  the  letters. 

Divide  the  first  term  of  the  dividend  by  the  first  term  of 
the  divisor ;  the  result  will  be  the  first  term  of  the  quotient. 

Multiply  the  whole  divisor  by  this  quotient,  and  subtract 
the  product  from  the  dividend. 

Consider  the  remainder  as  a  new  dividend,  and  proceed 
as  before  until  the  dividend  is  exhausted. 


DIVISION.  41 

Note. — If  the  dividend  is  not  exactly  divisible  by  the  divisor,  the 
remainder  must  be  placed  over  the  divisor  in  the  form  of  a  fraction 
and  connected  with  the  quotient  by  the  proper  sign. 

2.    Divide  a:^  +  4/  by  ar  —  2  ary  +  2/. 

ar^  -  2 xy  +  2f)  a:*  +  4^  {x'  J^2xy  +  2f 


2x'y- 
2a?y- 

-2x^f-\-4.y' 

2x^f  —  4.xy^ 
2  xV  — 4  a.  7/8 

Note.  —  By  multiplying  the  quotient  and  divisor  together  all  the 
terms  which  appear  in  the  process  of  dividing  will  be  found  in  the 
partial  products. 

3.    Divide  a:^  —  1  by  x  —  1. 

x—\)x^—l   (x^  +  ar^  +  ar  +  1 


x^ 

— 

•  1 

:^ 

— 

■x^ 

X?- 

1 

X?- 

X 

X 

— 

1 

X  ■ 

— 

1 

4.    Divide  rt  a?  —  ay-\-bx  —  by-\-zhyx  —  y 
X  —  y)  ax  —  ay  -\-  bx  —  by  -\-  z  {a  -\-  b  -\- 


x  —  y 
ax  —  ay 


bx  —  by 
bx  —  by 


5.    Divide  2by  —  2b' y  —  Zb-yz  -\-  Qb^y  -\-  byz  — yz 
by  2  b  —  z.  Ans.  Zb^y  —  by-\-y. 


42  ELEMENTARY   ALGEBRA. 

6.  Divide  d^  +  ^^  ^Y  c  -\-  x.  Ans.  c^  —  c x  -\-  x^. 

7.  Divide  a^-j-a^-{-a'x-^ax-^3ac-\-Sc  by  a  -|-  1. 

Ans.  a^  -{-  a  X  -\-  3  c, 

8.  Divide  a  -\-  b  —  d  —  ax  —  bx  -{-  dx  by  a  -\-  b  —  d. 

9.  Divide   2a*  —  ISa'ij -\-  11  a^f  —  S  ay" -{-  2y*   by 
2a^  —  ay-{-f.  Ans.  d^  —  6  ay  -^  2i/. 

10.  Divide  a^  —  SaH -\- 3  ab"^  —  b'^  by  a"^  —  2ab -\- i/. 

11.  Divide  2a^  —  19  x"  +  2Q  x  —  19  by  x  —  8. 
Ans.  2x'  —  Sx-\-  2—     ^ 


X  — 8. 

12.  Divide  ar'-f  1  by  a:  +  1. 

Ans.  X*  —  3^  -{-  x^  —  X  -\-  I. 

13.  Divide  x^  —  I  by  a:  —  1. 

Ans.  x«  +  X*  +  a:^  +  x^  +  X  +  1. 

14.  Divide  x^  —  1  by  x  -|-  1. 

15.  Divide  x^ — y^  by  x — y. 

Ans.  X*  4-  x'y  -|-  x^^^  +  xy  +  y. 

16.  Divide  a®  —  x^  by  a  —  x. 

17.  Divide  m^  —  n^  by  w^  +  ^  '^  +  '*^- 

Ans.  m*  —  m* «  -|-  m  «*  —  n*. 

18.  Divide  4a*  —  9a2_)_6a  —  3  by  2a^-\-3a—l. 

19.  Divide  x*  +  4x2y2  + 3y*  by  x  +  2y. 

20.  Divide  a*  —  a^x^ -f- 2ax' —  x<  by  or"  — ax  +  x^. 

21.  Divide  x«  +  2 x«y8  +  /  by  x^  —  xy  +  y\ 

22.  Divide  1  —  a*  by  1  +  «  +  a^  +  a\ 

Ans.   1  —  a. 

23.  Divide  10  x»  —  20  x^  ?/  +  30  y*  by  x  +  y. 

24.  Divide  7  a  x*  +  21  «  x»  -f  14  a  by  x  +  1. 

Ans.  7  rt  ar'  +  14  a  x^  —  14  a  x  +  14  a. 

25.  Divide  27  a*y  —  8  a»y  by  3 y'*  —  2  ay. 


DEMONSTKATION   OF   THEOREMS.  43 

SECTION   VIII. 

DEMONSTRATION    OF    THEOREMS. 

57.  From  the  principles  already  established  we  are  pre- 
pared to  demonstrate  the  following  theorems. 

THEOREM    I. 

The  sum  of  two  quantities  plus  their  difference  is  twice 
the  greater ;  and  the  sum  of  two  quantities  minus  their  dif- 
ference is  twice  the  less. 

Let  a  and  h  represent  the  two  quantities,  and  a  ^  6 ;  their  sum 
is  a  -\-h\  their  difference,  a  —  b. 

PROOF. 

1st.    (a4-J)4-(a_5)=«-[-5  +  a  —  6  —  2a; 
2d.     {a  +  h)  —  {a  —  h)=za-{-h  —  a-\-h  =  2b. 

Therefore,  when  the  sjim  and  difference  of  two  quanti- 
ties are  given  to  find  the  quantities, 

RULE. 

Subtract  the  difference  from  the  sum,  and  divide  the  re- 
mainder by  two,  and  we  shall  have  the  less ;  the  less  plus  the 
difference  will  be  the  greater. 

In  the  following  examples  the  sum  and  difference  are 
given  and  the  quantities  required. 

1.  16  and  12.  Ans.  14  and  2. 

2.  272  and  18. 

3.  456  and  84. 

4.  Sum  2  X  and  difference  2  y. 

Ans.  X -\- y  and  x  —  y. 

5.  Sum  1  x"^  -\-  St/  and  difference  b  x^  —  3  y. 

6.  Sum  2  a  —  8  6  and  difference  10  a  -[~  14  b. 

Ans.  6  a  -\-  3  b  and  —  4  a  —  11  b. 


44  ELEMENTARY   ALGEBRA. 

THEOREM    II. 
58i    The  square  of  the  sum  of  two  quantities  is  equal  to 
the  square  of  the  first,  plus  twice  the  product  of  the  two, 
plus  'the  square  of  the  second. 

Let  a  and  h  represent  the  two  quantities ;  their  sum  will  be 
a +  6. 

PROOF. 

a  +b 
a  +b 

a2+     ah 
+     ah  +  y" 

a2  _|.  2  a  i  +  62 
According  to  this  theorem,  find  the  square  of 

1.  a:  -|-  y.  Ans.  x^  -\'  2  xy  -\-  ij^. 

2.  2x  +  2y.  Ans.  4a;2-f  8xy +  4^^. 

3.  a:+  1. 

4.  4  +  a:. 

6.    2x-{-Sy.  AuB.  4:X^-\-12xy  +  9y\ 

6.    Sa-\-b. 

THEOREM    III. 

59*  The  square  of  the  difference  of  two  quantities  is 
equal  to  the  square  of  the  first,  minus  twice  the  product  of 
the  two,  plus  tlie  square  of  the  second. 

Let  a  and  b  represent  the  two  quantities,  and  a  ^  6 ;  their  dip 
ference  will  be  o  —  b. 

PROOF. 

a  —b 
a  —b 


a*  —     ab 

—     ab  +  b^ 


DEMONSTRATION  OF  THEOREMS.  45 

According  to  this  theorem,  find  the  square  of 

1.  X  —  y.  Ans.  x^  —  2xy-\-y'^. 

2.  2x  —  ^y. 

3.  x—l.  Ans.  x^  —  ^x+X. 

4.  7x  — 2. 

THEOREM    IV. 

60.   The  product  of  the  sum  and  difference  of  two  quan- 
tities is  equal  to  the  difference  of  their  squares. 

Let  a  -\-h  hQ  the  sum,   and   a  —  h  the  difference  of  the   two 
quantities  a  and  b. 

PROOF. 

a  +b 
a  —  b 
a^  +  ab 
—  ab-^I^ 

a^  —ly" 

According  to  this  theorem,  multiply 

1.  X  -\-y  hy  X  —  y.  Ans.  a;^  —  y^. 

2.  2  3:+  1  by  2  a:  —  1. 

3.  x'^  +  7/2  by  x^  —  y\  Ans.  x^  —  y\ 

4.  3  X  +  4  by  3  a;  —  4. 

6.    Sxy-^-'iabhyBxy  —  4a  5. 

61 .   This  theorem  suggests  an  easy  method  of  squaring 
numbers.     For,  since  a^  =  {a  —  b)  (a  -]-  ^)  +  ^^ 
992  =  (99  —  1)  (99  +  1)  +  12  =  98  X  100  +  1  =  9801. 
In  like  manner, 

962=    92  X     100  +  16  =  9216. 
9982  =  996  X  1000  +    4  =  996004. 
4972  =  494  X    500-1-    9  =  247  X  1000  +  9  =  247009. 


46  ELEMENTARY   ALGEBRA. 

In  accordance  with  this  principle  find  the  square  of 


1. 

98. 

4. 

493. 

7. 

888. 

2. 

89. 

5. 

789. 

8. 

999. 

3. 

45. 

6. 

698. 

9. 

1104. 

Miscellaneous  Examples. 

1.  Find  the  square  of  3  a:  —  6y. 

Ans.  dx'  —  BQxy  +  SQf. 

2.  Find  the  square  of4:axy-^7abx. 

Ans.   IGa'^x'' f+ 5QaHx''i/ +  4:9  aH'^x\ 

3.  Multiply  1  x-]-l  by  1  x  —  I.         Ans.  49  or^  —  1. 

4.  Required  those  two  quantities  whose  sum  is  3x  -\-2a 
and  difference  x  —  2  a.  Ans.  2  x  and  x  -\-  2  a. 

6.  Expand  (x^  —  ^y. 

6.  Multiply  4  a  />  +  3  by  4  a  i  —  3. 

7.  Find  the  square  of  14  a^  ^^  +  lO.ar^^. 

8.  Find  the  square  of  4  a  —  b. 

9.  Multiply  lOx  +  2  by  10a:  — 2. 

10.  Find  the  square  of  3  a  a:  —  9>axy. 

Ans.   ^a^s^  —  ^^a-cry-^-Ua^T^^. 

11.  Find  the  square  oi  2  a  -{-h. 

12.  Find  the  value  of  (6  a  +  4)  (6  a  —  4)  (36  a"-  +  16). 

Ans.  1296  a*  —  256. 

13.  Find  the  square  of  10  a^—  5  6^ 

14.  Expand  (3  a^a:  +  4  S/)2. 

Ans.   9  a^a:'^+ 24  aH a:/ -t- 16  6^V«. 

15.  Find  the  product  of  a^°  +  1,  a«  +  1,  a*  +  1,  a^  +  1, 
a  -\-  \,   and  a  —  1.  Ana.  a'^  —  1. 

16.  Find  the  product  of  a  -|-  ^»  ^  —  ^»  and  a^  —  i^. 


FACTORING.  47 

SECTION  IX. 

FACTORING. 

62.  Factoring  is  the  resolving  a  quantity  into  its  fac- 
tors. 

63.  The  factors  of  a  quantity  are  those  integral  quanti- 
ties whose  continued  product  is  the  quantity. 

Note.  —  In  using  the  word  factor  we  shall  exclude  unity. 

64.  A  Prime  Quantity  is  one  that  is  divisible  vt^ithout 
remainder  by  no  integral  quantity  except  itself  and  unity. 

Two  quantities  are  mutually  prime  when  they  have  no 
common  factor. 

65*  The  Prime  Factors  of  a  quantity  are  those  prime 
quantities  whose  continued  product  is  the  quantity. 

66.  The  factors  of  a  purely  algebraic  monomial  quan- 
tity are  apparent.  Thus,  the  factors  of  d^bxyz  are 
aXaXhXxXyXz. 

67.  Polynomials  are  factored  by  inspection,  in  accnrd- 
ance  with  the  principles  of  division  and  the  theorems  of 
the  preceding  section. 

CASE  I. 

68.  When  all  the  terms  have  a  common  factor. 

1.    Find  the  factors  of  ax  —  ah  -\-  ac. 

OPERATION.  As  a  is  a  factor  of 

(ax  —  ah  -\-  ac)  =  a  (x  —  h  -\-  c)  each  term  it  must  be 

a  factor  of  the  poly- 
nomial ;  and  if  we  divide  the  polynomial  by  a,  we  obtain  the  other 
factor.     Hence, 


48  ELKMENTARY   ALGEBRA. 

RULE. 
Write  (he  quotient  of  the  polynomial  divided  by  the  com- 
vion  factor  in  a  parenthesis,  with  the  common  factor  lyre- 
fixed  as  a  coefficient. 

2.  Find  the  factors  of6x9/—12xf-{-lSax^i/\ 

Ans.  6x^^(1  —  12^/  + 3ax/). 

Note.  —  Any  factor  common  to  all  the  terms  can  be  taken  as  well 
as  6  a;y ;  2,  3,  ar,  y,  or  the  product  of  any  two  or  more  of  these  quan- 
tities, according  to  the  result  which  is  desired.  In  the  examples 
given,  let  the  greatest  monomial  factor  be  taken. 

3.  Find  the  factors  of  a;  +  x^.  Ans.  x  {I  -\-  x). 

4.  Find  the  factors  of  S  a^  x"^  -\-  12  a^  x*  —  ^axy. 

Ans.  4  a  x  (2  a  x  +  Zo^  3?  —  y). 

6.  Find  the  factors  oi  h  x"  y'' -\- "Ih  a  x''  —  15  ar^/. 

Ans.   5a:^(x/+ 5aa:2  — 3/). 

6.  Find  the  factors  of  ^  ax  —  8  ^^  -|-  14 a;^ 

T.  Find  the  factors  of  4  x^y^  —  28  x^y  —  44  x^  y^ 

8.  Find  the  factors  of  55  a^  c  —  11  a  c  +  33  a*  c  x. 

9.  Find  the  factors  of  98  a^a;^  —  294  a«x2y2 

10.  Find  the  factors  o{  Ihd'l?  cd —  ^  aWd'' -{-X'^a^^d^. 

CASE    II. 
69.   When  two  terms  of  a  trinomial  are  perfect  squares 
and  positive,   and  the   third   term   is   equal   to  twice   the 
product  of  their  square  roots. 

1.  Find  the  factors  of  a^  +  2  a  6  +  ^2. 

OPERATION.  WTe  rcsolve  this  into 

c^  ■\-  1  ah  -^  h^  z=z  {a  -\-  h)  {a  -\-  h)  its  factors  at  once   by 

the    converse     of    the 
principle  in  Theorem  II.  Art.  58. 


FACTORING.  49 

2.  Find  the  factors  of  a^  _  2  a  6  +  61 

OPERATION.  We  ^esol^e  tl^ig  j^to 

rt^  —  2  a  6  -|-  6^  =  (a  —  b)  (a  —  b)  its  factoi-s   at  once  by 

the    converse     of    the 
principle  in  Theorem  III.  Art.  59.     Hence, 

RULE. 
Omilting  the   term  that  is  equal  to  twice  the  product  of 
the  square  roots  of  the  other  two,  take  for  each  factor  the 
square  root  of  each  of  the  other  two  connected  by  the  sign 
of  the  term  omitted. 

3.  Find  the  factors  of  ^r^  —  2xy  -{-  1/^. 

Ans.   {x—y)  {x  —  y), 

4.  Find  the  factors  of  4  a^  c2  -f-  12  a  c  cZ  +  9  d\ 

Ans.   (2  a  c  +  3  rf)  (2  a  c  +  3  rf). 
6.  Find  the  factors  of  1 — A:xz  -\-  ^3?  z^. 

Ans.   {\  —2xz)  {l—2xz). 
6.  Find  the  factors  of  9  a;^  —  6  x-  +  1. 

Ans.   (3  a;  — 1)  (3ar—  I). 
T.  Find  the  factors  of  25  x^  +  60  x-  +  36. 

8.  Find  the  factors  of  49  a^  —  14  ax  -[-  ^^^ 

Ans.  (la  —  x)  {la  —  x). 

9.  Find  the  factors  of  16  y^  —  iQa'y  +  4  a*. 

10.  Find  the  factors  of  12  ax  +  4:X^  +  9  a"^, 

11.  Find  the  factors  of  6  x  +  1  +  9  a^. 

CASE    III. 
70.    When   a  binomial   is  the    difference   between    two 
squares. 

1.  Find  the  factors  of  a^  —  5^ 

OPERATION.  We  resolve  this  into  its  fac- 

2 i^ /      I    ^\  /    ^\  tors  at  once  by  the  converse  of 

the  principle  in  Theorem  IV. 
Art.  60.     Hence, 


50  KLEMENTARY  ALGEBRA. 

RULE. 

Take  for  one  of  the  factors  the  sum,  and  for  the  other 
the  difference,  of  the  square  roots  of  the  terms  of  the  bi- 
nomial. 

2.  Find  the  factors  of  x^  —  f. 

Ans.   (x  +  1/)  {x-^y). 

3.  Find  the  factors  of  4  a^  —  9  i*. 

Ans.  (2a  +  3  62)  (2a  — 3^). 

4.  Find  the  factors  of  16  x^  —  c^. 

5.  Find  the  factors  of  o^  6*6'^  —  x^y'^. 

6.  Find  the  factors  of  81  x"^  —  49  3/^. 
T.  Find  the  factors  of  25  0^  —  4  t*. 

8.  Find  the  factors  of  m^  —  n^^. 

Note.  —  When  the  exponents  of  each  term  of  the  residual  factor 
obtained  by  this  rule  are  even,  this  factor  can  be  resolved  again  by 
the  same  rule.  Thus,  x*  —  3/*  =  (^  -h  V^)  (^  —  y^)  \  but  a:*  —  y*  = 
(a;  -f  y)  {x  —  y)\  and  therefore  the  factors  of  x*  —  /  are  x'  -|-  y*, 
X  -\-  y^  and  x  —  y. 

9.  Find  the  factors  of  a^  —  h\ 

Ans.    (a^-^lf)  {a-\-h)  (a  —  b). 

10.  Find  the  factors  of  a:^ — y^. 

Ans.  {x'  +  y')  (^'  +  /)  (^  +  y){^-  y)' 

11.  Find  the  factors  of  a*  —  1. 

12.  Find  the  factors  of  1  —  x\ 

Ans.    (1  +  x')  (1  +  x")  (1  +  x)  (1  —  ar). 

13.  Find  the  factors  of  a'  —  a^. 

Ans.  a^(a-\-  1)  {ii  —  1). 

14.  Find  three  factors  of  x^  —  a?. 

71.  Any  binomial  consisting  of  the  difference  of  the 
same  powers  of  two  quantities,  or  the  sum  of  the  same 
odd  powers,  can  be  factored.     For 


FACTORING.  51 

I.    The  difference  of  the  same  powers  of  two  quantities 
is  divisible  by  the  difference  of  the  quantities. 

Let  a  and  b  represent  two  quantities  and  a^b,  and  by  actual 
division  we  find 

(T  —  b"' 


—  a-\-b, 

—  a'j^ab-^y^, 

a  —  b  '  '  ' 


a  —  b 

a  —b 
a*  —  b' 


and  so  on. 

II.  The  difference  of  the  same  even  powers  of  two  quan- 
tities is  divisible  by  the  sum  of  the  quantities. 

a^  —  b'' 


a  +6 

a'  —  b* 
a  +6 


z=.  a  —  b, 

—  a^  —  an  +  ab^  —  W, 

=  a''  —  a'b-\-aH^  —  a'  h^  +  a  &*  —  b\ 


and  so  on. 

It  follows  from  the  two  preceding  statements  that 

The  difference  of  the  same  even  powers  of  two  quantities 
is  divisible  by  either  the  sum  or  the  difference  of  the  quan- 
tities. 

III.  The  sum  of  the  same  odd  powers  of  two  quantities  is 
divisible  by  the  sum  of  the  quantities. 

a  -\-b  ' 

"^-i^  =:a'  —  an  +  aH'  —  ab'+  b\ 
a  ■\-o  ' 

and  so  on. 


52  KLEMENTARY    ALGEBRA. 

1.  Find  the  factors  of  ar'  —  y*. 

OPERATION. 

{^  -y')  -^  {^^  -y)  =x'  +  x'y  +  x^f  +  xf  +  y^ 

By  I.  of  this  article,  the  difference  of  the  same  powers  of  two 
quantities  is  divisible  by  the  dilTerence  of  the  quantities;  therefore 
X  —  y  must  be  a  factor  ofV  —  if-^  and  dividing  or*  —  y*  by  z — y 
gives  the  other  factor  x*  -\-  x^  y  -\-  x"'  y"-  -\-  x  if  -\-  y. 

2.  Find  two  factors  of  c^  —  d\ 

OPERATION. 

{c^  —  d^)  ^  {c-^d)  =c^-^c^d  +  c'd^  —  c''d^-\-cd^  —  d'' 

By  II.  the  difference  of  the  same  even  powers  of  two  quantities 
is  divisible  by  the  sum  of  the  quantities ;  therefore  c  -j-  d  must  be  a 
fiictor  of  c*  —  J^;  and  dividing  c'  —  d^  by  c  -\-  d  gives  the  other 
factor  c'^  —  c*  (/  +  c'  d^  —  c'  d^ -\.  c  d' —  d\ 

3.  Find  the  factors  of  m^  -\-  n^. 

OPERATION. 

(m«  +  n^)  -r-  (m  +  w)  =  m*  —  w^  n  -[-  m^  n*  —  7n  w^  _|_  n* 

By  III.  the  sum  of  the  same  odd  powers  of  two  quantities  is 
divisible  by  the  sum  of  the  quantities ;  therefore  m -\-  n  must  be  a 
factor  of  m'  -|-  n^ ;  and  dividing  m'  -\-  n^  hy  m -\-  n  gives  the  other 
factor  m*  —  n^  n  -\-  rr^  r?  —  mr?  -\-  n*. 

4.  Find  the  factors  of  c^  —  di?. 

Ans.   (a  —  x)  {c? -^  a  X -\- x^) , 

5.  Find  the  factors  of  a^  +  ^'^• 

Note.  —  In  Example  2,  the  factors  of  c"  —  </•  there  obtained  are 
not  the  only  factors;  for  by  I.  c*  —  rf*  is  divisible  by  c  —  d\  and 
dividing  c*  —  d'  by  c  —  d  gives  another  factor, 

c*  4-  c*  rf  4-  c«  ^/»  +  c=  (/^  4-  c  rf*  -f  d» ; 
or  by  Art.  70, 

c«  —  ^^  =  (c«  +  c/')  (c»  —  d»). 


FACTORING.  53 

But  c'  —  f"  d  -\- c""  d^  —  c"  d^ -\- c  d^  —  d\ 

c^  _|_  c*  (I -{- c' d"" -\- (f  d' -{- c  d* -\-  d\ 
e  +  d\ 
c'  —  d^ 

are  not  prime  quantities ;  for  the  first  can  be  divided  by  c  —  d,  and 
the  quotient  thus  arising  can  be  divided  by  c^  ±  c  d  -\-  d"^;  the  second 
can  be  divided  by  c  -|-  ^>  ^^^  the  quotient  thus  arising  will  be  the 
same  as  after  the  division  of  the  first  quantity  by  c  —  d,  and  can 
be  divided  hy  (^  ±  c  d  -\-  d^;  the  third  can  be  divided  by  c  -f-  ^j  and 
the  fourth  by  c  —  d.  Performing  these  divisions,  by  each  method 
we  shall  find  the  prime  factors  of  c®  —  d®  to  be 

c-\-d,  c  —  d,  c-  +  c  (/  +  rP,  and  c^  —  c  </  -f  d\ 

In  finding  the  prime  factors,  it  is  better  to  apply  first  the  princi- 
ple of  Art.  70  as  far  as  possible. 

6.  Find  the  prime  factors  of  x^^  —  y^^. 

x^  +y'  ={x+  y)  {x^  —  a^y-^-x'y^  —  xf-^-  y^). 
x^  —y^   ={x  —  y)  (x*  -\-^y-\-  ^f  -\-xf  -\-  y^). 

Ans.   (x-\-y)(x—y)  {x^  —  x^y-\-x'y'^  —  xf 

+  y')  {^'  +  ^'y  +  ^/  +  ^y  +  y')' 

T.  Find  the  prime  factors  of  a^  —  1. 

Ans.   («  +  l)  («— 1)  («'  +  «+!)  (a^  —  a+l). 

8.  Find  the  prime  factors  of  a^  —  2  d^  x^  +  x^. 

Ans.   {a  -\-x)  {a -\-  x)  {a  —  x)  {a  —  x). 

9.  Find  the  prime  factors  of  x^  -\-  2x^f-\-y^. 

Ans.   {x-\-y)  {x-\-y)  {^  —  xy^f)  {x^  —  xy  +  f). 

10.  Find  the  prime  factors  of  1  —  a^. 

Ans.   (1  +«)(!—  a)  (1  +  a"). 

11.  Find  the  prime  factors  of  8  —  c^. 

Ans.   (2  — c)  (4  +  2c  +  c2). 


ELEMENTARY  ALGEBRA. 


SECTION   X. 

GREATEST    COMMON    DIVISOR.* 

72.  A  Common  Divisor  of  two  or  more  quantities  is  any 
quantity  that  will  divide  each  of  them  without  remainder. 

73.  The  Greatest  Common  Divisor  of  two  or  more 
quantities  is  the  greatest  quantity  that  will  divide  each 
of  them  without  remainder, 

74.  To  deduce  a  rule  for  finding  the  greatest  common 
divisor  of  two  or  more  quantities,  we  demonstrate  the 
two  following  theorems  :  — 

Theorem  I.  A  common  divisor  of  two  quantities  is  also 
a  common  divisor  of  the  sum  or  the  difference  of  any 
multiples  of  each. 

Let  A  and  B  be  two  quantities,  and  let  d  be  their  common  di- 
visor ;  d  is  also  a  common  divisor  of  m  ^  ±  n  5. 

Suppose  A  -^  d  ^=  p\  i.  e.  A  =  dp,  and  m  A  =  dmp, 
and  B  ^  d  =  q'^  i.  e.  B  =^  d q,   and  n  B  ^:=  d  n  q\ 

then  mA  ±  nB  =  dmp±dnq=d{mp±  nq). 

That  is,  d  is  contained  in  m  A  -{-  n  B,  m  p  -{-  n  q  times,  and  in 
m  A  —  n  B,  m  p  —  n  q  times ;  i.  e.  </  is  a  common  divisor  of  the 
sum  or  the  difference  of  any  multiples  of  A  and  B. 

Theorem  II.  The  greatest  common  divisor  of  two  quan- 
tities is  also  the  greatest  common  divisor  of  the  less  and 
the  remainder  after  dividing  the  greater  by  the  less. 

Let  A  and  B  be  two  quantities,  and  yl  >  J5 ; 
and  let  the  process  of  dividing  be  as  appeai-s  in  B)    A  (q 

the  margin.     Then,  as  the  dividend  is  equal  to  qB 

the  product  of  the  divisor  by  the  quotient  plus  ~ 

the  remainder, 

A=r+qB.  (1) 

•  See  Prefiwe. 


GREATEST    COMMON   DIVISOR.  55 

And,  as  the  remainder  is  equal  to  the  dividend  minus  the  product 
of  the  divisor  by  the  quotient, 

r  =  A—qB.  (2) 

Therefore,  according  to  the  preceding  theorem,  from  (1)  any  divisor 
of  r  and  B  must  be  a  divisor  of  .4  ;  and  from  (2)  any  divisor  of  A 
and  B,  a  divisor  of  r ;  i.e.  the  divisors  of  A  and  B  and  B  and  r 
are  identical,  and  therefore  the  greatest  common  divisor  of  A  and 
B  must  also  be  the  greatest  common  divisor  of  B  and  r. 

In  the  same  way  the  greatest  common  divisor  of  B  and  r  is  the 
greatest  common  divisor  of  r  and  the  remainder  after  dividing  B 
by  r. 

Hence,  to  find  the  greatest  common  divisor  of  any  two 
quantities, 

RULE. 

Divide  the  greater  by  the  less,  and  the  less  by  the  remain- 
der, and  so  continue  till  the  remainder  is  zero;  the  last  di- 
visor is  the  divisor  sought. 

Note  1.  —  The  division  by  each  divisor  should  be  continued  until 
the  remainder  will  contain  it  no  longer. 

Note  2.  —  If  the  greatest  common  divisor  of  more  than  two  quan- 
tities is  required,  find  the  greatest  common  divisor  of  two  of  them, 
then  of  this  divisor  and  a  third,  and  so  on ;  the  last  divisor  will 
be  the  divisor  sought. 

Note  3.  —  The  common  divisor  of  xy  and  xzia  x;  x  is  also  the 
common  divisor  of  x  and  x z,  or  of  « a: y  and  xz;  i.  e.  the  common 
divisor  of  two  quantities  is  not  changed  by  rejecting  or  introducing 
into  either  any  factor  which  contains  no  factor  of  the  other. 

Note  4.  —  It  is  evident  that  the  greatest  common  divisor  of  two 
quantities  contains  all  the  factors  common  to  the  quantities. 

CASE    I. 
75.    To  find  the  greatest  common  divisor  of  monomials. 
1.    Find    the    greatest    common    divisor    of    S  a^  b^  c  d, 
lQaH'c\  and  2S  aH^  c. 

The  greatest  common  divisor  of  the  coefficients  found  by  the  gen- 
eral rule  is  4  ;  it  is  evident  that  no  higher  power  of  a  than  a^,  of 


56  ELEMENTARY  ALGEBRA. 

b  than  I^,  of  c  than  itself,  will  divide  the  quantities ;  and  that  d  will 
not  divide  them ;  therefore,  the  divisor  sought  is  4  a' t^  c.     Hence, 

RULE. 

Annex  to  the  greatest  common  divisor  of  the  coefficients 
fJwse  letters  which  are  comm,on  to  all  the  quantities,  giving  to 
each  letter  the  least  exponent  it  has  in  any  of  the  quantities. 

2.  Find  the  greatest  common  divisor  of  63  a*  U^  c*  d^, 
2T  a^  h^  c^  and  45  a^  h^  c^  d.  Ans.  9  a^  IP  c\ 

3.  Find  the  greatest  common  divisor  of  Ibx'y^s^  and 
125  a  bx^fz\ 

4.  Find  the  greatest  common  divisor  of  99  a  b^  c*  d^  x^  i/^ 
and  22  a' b^c^d'^x^.  Ans.   11  a  ly^c^d^x^. 

6.  Find  the  greatest  common  divisor  of  11  x^y^,  \9x^y^, 
and  2\2bx'y^z^. 

CASE   II. 
76»   To  find  the  greatest  common  divisor  of  polynomials. 

1 .  Find  the  greatest  common  divisor  of  x"^  —  y'^  and 
x^  —  2xy  -\-  rf-. 

■x'-f)x^-r.2xy-\-    /(I 


x^ 

—  .r 

^2xy  +  2y' 

Eejectmg 

the  factor 2y 

X-   y)x^- 

-xy 

+  y 

xy- 
xy- 

Ans. 

X  - 

-V' 

RULE. 

Arrange  the  terms  of  both  quantities  in  the  order  of  the 
poivers  of  some  letter,  and  then  proceed  according  to  the 
general  rule  in  Art.  74. 

Note  1.  —  If  the  leading  term  of  the  dividend  is  not  divisible  by 
the  leading  term  of  tlio  divisor,  it  can  be  made  so  by  introducing 


GREATEST   COMMON   DIVISOR.  57 

in  the  dividend  a  factor  which  contains  no  factor  of  the  divisor ;  or 
either  quantity  may  be  simplified  by  rejecting  any  factor  which 
contains  no  factor  of  the  other.     (Art.   74,  Note  3.) 

Note  2.  —  Since  any  quantity  which  will  divide  a  will  divide  —  a, 
and  vice  versa,  and  any  quantity  divisible  by  a  is  divisible  by  —  a, 
and  vice  versa,  therefore  all  the  signs  of  either  divisor  or  dividend, 
or  of  both,  may  be  changed  from  -|-  to  — ,  or  —  to  -J-,  without 
changing  the  common  divisor. 

Note  3.  —  When  one  of  the  quantities  is  a  monomial,  and  the  other 
a  polynomial,  either  of  the  given  rules  can  be  applied,  although  gen- 
erally the  greatest  common  divisor  will  be  at  once  apparent. 

2.  Find  the  greatest  common  divisor  of  ax'  —  a'^x'*  —  8  a^x^ 
and  2  c  x^  —  2  a  c  x^  -\-  4.  a^  c  x^  —  6  a^  c  X  —  20  a'^  c. 

ax'  —  a^x^  —  8 a'a;^]  2 ex*—  2a ca:^  +  4 a- ex"-  —  6 a^ca:—  20 a^c 

Dividing  by  2  C 

8a*      )X^  —  a3(?-\-2a''x'''  —  Za'x—l0a^  {I 
x^  —  ax?  —    8  a* 


2a2x2  — Sa'a;  — 2a* 

Dividing  by  O 

2x2  — 3  ax— 2  a^ 


2  a'^  X-  —  3  a^  X  —    2  a*  ist  Rem. 
x^—    ax*—    8  a* 

Multiplying  by    2 

2x*— 2rtx^— 16a*  (x2 

2x*  — 3ax^  — 2  0^x2 

ax^+2a2x-— 16a* 

Multiplying  by  2 

2ax'  +  4a2x'^  — 32a*  (ax 

2ax^  — 3a'x^—    2a^x 

la^x^-\-    2a^x  — 32a* 

Multiplying  by  2 

Ua^x^-f-    4a'x  — 64a*  (Za'* 
14a'^x''— 21a^x— 14  a* 
25  a' X  — 50  a*  25a^x  — 50  a*  2d  Rem. 

Dividing  by  25  a' 

a;_2a)  2x'  — 3  ax— 2  0^(2  X -fa 
2  x^  —  4  a  X 

ax  —  2  a^ 

ax  —  2 a^  Ans.  x  —  2 a. 

3* 


58  ELEMENTARY    ALGEBRA. 

3.  Find   the    greatest   common    divisor    of  a*  —  x*   and 

4.  Find   the    greatest    common    divisor  of  a*  —  x*   and 
a^  _  a^  .x2.  Ans.  a^  —  x^. 

5.  Find  the  greatest  common  divisor  ot'2ax^  —  a^x  —  a' 
and  2  x'^ -\- Z  a  X -\- d\ 

6.  Find   the   greatest   common   divisor   of   Q  a  x  —  8  a 
and  6  a  ic^  + a  or-— 12  a  a:.  Ans.  3  a  a:  — 4  a. 

T.  Find    the  greatest   common    divisor   of  x*  —  y*   and 
x'  +  f. 

8.  Find  the  greatest  common  divisor  of  3  a:*  —  2i:X  —  9 
and  2a:^—  16x  — 6. 

9.  Find   the  greatest  common   divisor  of  x^  —  y^   and 
x'^  —  y"^.  Ans.  x  —  y. 

10.  Find  the  greatest  common  divisor  of  \Qx^  —  20x^y 
+  30  /  and  x^  -{- 2x^y  +  2  xy'^  +  y\  Ans.  x  +  y.   . 

11.  Find  the  greatest  common  divisor  of  a* -\- a^ -\- a^ 
+  a  —  4  and  a^  +  2  a»  +  3  a^  +  4  a  —  10. 

Ans.  a  —  1. 

12.  Find  the  greatest  common  divisor  of  1  ax*  -\-  2lax^ 
-f  14  a  and  x^  -\-  x*  -\-  x^  —  x.  Ans.  x  -\-  I. 

13.  Find  the  greatest  common  divisor  of  27  a^ y*  —  Sa^y 

and  3  y  —  2  a  /  +  3  a^/  —  2  a«y^ 

Ans.  3y^  —  2  ay. 

14.  Find  the  greatest  common  divisor  of  n^  -\-  a  —  10 
and  a*  —  16.  Ans.  a  —  2. 

Note  5.  —  The  greatest  common  divisor  of  polynomials  can  also  be 
found  by  factoring  the  polynomials,  and  finding  the  product  of  the 
factors  common  to  the  polynomials,  taking  each  factor  the  least  num- 
ber of  times  it  occurs  in  any  of  the  quantities.     (Art.  74,  Note  4.) 

15.  Find  the  greatest  common  divisor  of  3  a  j:^  —  4oj:  -|- 
Saxy  —  4:  ay  and  a^ x  —  x  -\-  a^ y  —  y. 

3  ax^  —  4rax-\-3axy  —  4a?/  =  a{x  +  y)  (3  j;  —  4) 
a^r  —  X  -\-  a'^  y  ~  y=  (x-\-y)  {a  —  1)  (rt-^  +  a  +  0 

Ans.  X  -f-  //. 


LEAST   COMMON   MULTIPLE.  59 

SECTION   XI. 

LEAST    COMMON    MULTIPLE. 

77.  A  Multiple  of  any  quantity  is  a  quantity  that  can 
be  divided  by  it  without  remainder. 

78.  A  Common  Multiple  of  two  or  more  quantities  is 
any  quantity  that  can  be  divided  by  each  of  them  with- 
out remainder. 

79.  The  Least  Common  Multiple  of  two  or  more  quan- 
tities is  the  least  quantity  that  can  be  divided  by  each 
of  them  without  remainder. 

80.  It  is  evident  that  a  multiple  of  an}''  quantity  must 
contain  the  factors  of  that  quantity ;  and,  vice  versa,  any 
quantity  that  contains  the  factors  of  another  quantity  is 
a  multiple  of  it :  and  a  common  multiple  of  two  or  more 
quantities  must  contain  the  factors  of  these  quantities  ; 
and  the  least  common  multiple  of  two  or  more  quantities 
must  contain  only  the  factors  of  these  quantities. 

CASE  I. 
To  find  the  least  common  multiple  of  monomials. 

1.  Find  the  least  common  multiple  of  6  a^h^c,  8  a^¥  c^df 
and  \2a^bcx. 

The  least  common  multiple  of  the  coefficients,  found  by  inspection 
or  the  rule  in  Arithmetic,  is  24 ;  it  is  evident  that  no  quantity  which 
contains  a  power  of  a  less  than  a*,  of  b  less  than  i*,  of  c  less  than 
C-,  and  which  does  not  contain  d  and  a;,  can  be  divided  by  each  of 
these  quantities  ;  therefore  the  multiple  sought  is  24  a*  h^  c^  d  x. 

Hence,  in  the  case  of  monomials, 


60  ELEMENTARY   ALGEBRA. 

RULE. 

Annex  to  the  least  common  multiple  of  the  coefficients  all 
the  letters  which  appear  in  the  several  quantities,  giving  to 
each  letter  the  greatest  exponent  it  has  in  any  of  the  quan- 
tities. 

2.  Find  the  least  common  multiple  of  3  a*  IP-  c^  ^  a!  h'' c  d"^, 
and  l^ahcx^.  Ans.  Z^  a!  h^  c^  d"^  i^ . 

3.  Find  the  least  common  multiple  of  X^ahx,  80ai*ar^ 
and  Zha^hxK  Ans.   hm  a^  I/' x\ 

4.  Find  the  least  common  multiple  of  9a'6^  Xha^hx^, 
and  X'^axf.  Ans.   ^^a^h^x^y^. 

6.  Find  the  least  common  multiple  of  ISa'^c^a:, 
l^aVcx'y,  and  Z^d'y'xz. 

6.  Find  the  least  common  multiple  of  X^^xyz,  4:5  a  be, 
and  25  m  n. 

*l.  Find  the  least  common  multiple  of  10  a'^hy'^,  13  a*  IP  c, 
and  ITa^&s'c^ 

8.  Find  the  least  common  multiple  of  14  a^  IP  c*,  20  a^  h  c^, 
2baHc^,  and  2S  abed. 


CASE  II. 

81.  To  find  the  least  common  multiple  of  any  two 
quantities. 

Since  the  greatest  common  divisor  of  two  quantities  contains  all 
the  factors  common  to  these  quantities  (Art.  74,  Note  4)  ;  and  since 
the  least  common  multiple  of  two  quantities  must  contain  only  the 
factors  of  these  quantities  (Art.  80)  ;  if  the  product  of  two  quanti- 
ties is  divided  by  their  greatest  common  divisor,  the  quotient  will 
be  their  least  common  multiple. 

Hence,  to  find  the  least  common  multiple  of  any  two 
quantities, 


LEAST   COMMON  MULTIPLE.  61 

RULE.  ^ 

Divide  one  of  the  quantities  by  their  greatest  common  di- 
msor,  and  multiply  this  quotient  by  the  other  quantity,  and 
the  product  will  be  their  least  common  multiple. 

Note  1.  —  If  the  least  common  multiple  of  more  than  two  quanti- 
ties is  required,  find  the  least  common  multiple  of  two  of  them, 
then  of  this  common  multiple  and  a  third,  and  so  on ;  the  last  com- 
mon multiple  will  be  the  multiple  sought. 

Note  2.  —  In  case  the  least  common  multiple  of  several  monomials 
and  polynomials  is  required,  it  may  be  better  to  find  the  least  com- 
mon multiple  of  the  monomials  by  the  Rule  in  Case  I.,  and  of  the 
polynomials  by  the  Rule  in  Case  11.,  and  then  the  least  common 
multiple  of  these  two  multiples  by  the  latter  Rule. 

1.  Find    the    least    common    multiple    of   x^  —  y^    and 

OPERATION.  Their  greatest  common 

X  —  y)  x^  —  2x1/  -\-y^  divisor   is    x  —  y,   with 

: which  we  divide  one  of 

•^        y  'the  quantities ;  and  mul- 

{x^  —y"^)  (x^y),  Ans.  tiplying  the  other  quan- 
tity by  this  quotient,  we 

have  the  least  common  multiple  (j^  —  y^)  (x  —  y). 

2.  Find  the  least  common  multiple  of  2a^x'^,  4cX^yt 
a^  —  X*,  and  a^  —  a^  x^. 

The  least  common  multiple  of  the  monomials  is  ^c?  x^y\  and 
the  least  common  multiple  of  the  polynomials  is  c?  (a*  —  x*). 

The  greatest  common  divisor  of  these  two  multiples  is  c? ;  and 
dividing  one  of  these  multiples  by  a'^,  and  multiplying  the  quotient 
by  the  other,  we  have  4  a^  a:^  y  («*  —  x*)  as  the  least  common  mul- 
tiple. 

3.  Find  the  least  common  multiple  of  3  a^  h^,  6  a^by, 
a»  —  8,  and  a^  —  4:a-{-  4. 

Ans.   6  a^  b^y  {a^  —  8)  (a  —  2). 


62  ELKMENTAKY    ALGEBRA. 

4.  Find  the  leust  common  multiple  of  3  x^  —  24  ar  —  9 
and  2x^—l6x  —  Q. 

(See  8th  Example,  Art.  76.) 

5.  Find  the  least  common  multiple  of  a*  —  x*  and 
a^  —  x^. 

6.  Find  the  least  common  multiple  of  a:* —  l,x^-\-2x  -{-  I, 
and  {x  —  ly.  Ans.  x""  —  x^  —  x^  +  I. 

7.  Find  the  least  common  multiple  of  x*  —  y*  and  x^  -(-  y. 

8.  Find  the  least  common  multiple  of  a^  -|-  a  —  10  and 
a'  —  16. 

Note  3.  —  The  least  common  multiple  of  any  quantities  can  also 
be  found  by  factoring  the  quantities,  and  finding  the  product  of  all 
the  factors  of  the  quantities,  taking  each  factor  the  greatest  number 
of  times  it  occurs  in  any  of  the  quantities.     (Art  80.) 

9.  Find  the  least  common  multiple  of  x^  —  2xy-\-y^, 
X*  —  y*,  and  {x  -\-  yy. 

x^—  2xy  +  y^={x  ~y)  (x  —  y) 

x^  —  y^=  (j?2  -|-  if)  {x  J^2j)  {x~  y) 
(x  +  yy  ={x-\-y)  {x  +  y) 
Hence  L.  C.  M  =  {x  —  y)  {x~y)  {x'^  +  y'')  {x-\-y)  (ar  +  y) 
=  x''  —  x^y''  —  x'^y'-\-y\ 

10    Find  the  least  common  multiple  of  3a a;*  —  4aar-|- 
S  axy  —  4:ay  and  a^ x  —  x  -\-  a^y  —  y. 
(See  15th  Example,  Art.  76.) 
Ans.  o(a?  +  ?/)  (3^7  — 4)  (a^+a  +  l)  (a  — 1)  =:3a*a:2_ 

4:a*x  -\-  Sa*xy  —  4a^i/  —  Sax^  -["  ^^^  —  Saxy  +  ^ay. 


FKACTIONS.  68 

SECTION   XII. 

FRACTIONS. 

82.  When  division  is  expressed  by  writing  the  dividend 
over  the  divisor  with  a  line  between,  the  expression  is 
called  a  Fraction.  As  a  fraction,  the  dividend  is  called 
the  numerator,  and  the  divisor  the  denominator. 

Hence,  the  value  of  a  fraction  is  the  quotient  arising 
from  dividing  the  numerator  by  the  denominator. 

XV 

Thus,  is  a  fraction  whose  numerator  is  x  y  and  denominator  y^ 
and  whose  value  is  x. 

83.  The  principles  upon  which  the  operations  in  frac- 
tions are  carried  on  are  included  in  the  following 

THEOREM. 
Any  multiplication  or  division  of  the  numerator  causes  a 
like  change  in  the  value  of  the  fraction,  and  any  multiplica- 
tion or  division  of  the  denominator  causes  an  opposite  change 
in  the  value  of  the  fraction. 

X  v^ 
Let  — ^   be  any  fraction :  its  value  =  xy. 
y  '  ' 

1st.   Changing  the  numerator. 

Multiplying  the  numerator  by  y, 

which  is  y  times  the  value  of  the  given  fraction. 
Dividing  the  numerator  by  y, 

y 

which  is  —  of  the  value  of  the   given  fraction. 


64  ELEMENTARY   ALGEBRA. 

2d.    Changing  the  denominator. 

Multiplying  the  denominator  by  y, 

f 

which  is  -  of  the  value  of  the  given  fraction. 
Dividing  the  denominator  by  y, 

which  is  y  times  the  value  of  the  given  fraction. 


Corollary.  — Multiplying  or  dividing  both  numerator  and 
denominator  by  the  same  quantity  does  not  change  the  value 
of  the  fraction. 

For  if  any  quantity  is  both  multiplied  and  divided  by  the  same 
quantity  its  value  is  not  changed. 

rr.,         a:  y         c xy         x 
Thus,  — ^  =  — -  =  -   =  r. 
y  cy  1 

84.  Every  fraction  has  three  signs:  one  for  the  numer- 
ator, one  for  the  denominator,  and  one  for  the  fraction 
as  a  whole. 

Thus,  +=^^ 

—  0 

If  an  even  number  of  these  signs  is  changed  from  -\-  to 
— ,  or  —  to  +>  ih^  value  of  the  fraction  is  not  changed ; 
but  if  an  odd  number  is  changed,  the  value  of  the  fraction 
is  changed  from  -\-  to  — ,  or  —  to  -\-. 

Thus,  changing  an  even  number, 

-f-a:y  _  __  —  xy  _  _  -f-ary  _    ,    —xy  _    .    ^ . 

but,  taking 


FRACTIONS.  65 

and  changing  an  odd  number, 

_  +  ^y  _  I  —JLU  —  _!_  +  ^y _.  _  —  ^y  ^=  —  x 
-\-y        "^  -^ y  —y  —y 

The  various  operations  in  fractions  are  presented  under 
the  following  cases. 

CASE    I. 
85.    To  reduce  a  fraction  to  its  lowest  terms. 

Note.  —  A  fraction  is  in  its  lowest  terms  when  its  terms  are  mu- 
tually prime. 

1.  Reduce  ^,—o—o  to  its  lowest  terras. 

OPERATION.  Since  dividing  both  terms 

Ua'xy    _Axy    __     2  ^^  ^  ^'^^^^°"  ^^  ^^'^  '^"'« 

2T^"p  —  6^  —  Vy  quantity   does   not   change 

its  value  (Art.  83,  Cor.),  we 
divide  both  terms  by  any  factor  common  to  them,  as  4  a^ ;  and  both 
terms  of  the   resulting  fraction   by  any  factor  common  to  them,  as 

2xy\  or  we  can  divide  both  terms  of  the  given  fraction  by  their 

2 
greatest  common  divisor  ^cPxy\  the  resulting  fraction  —   is  the 

fraction  sought.     Hence, 

RULE. 
Divide  both  terms  of  the  fraction  by  any  factor  common 
to  them ;  then  divide  these  quotients  by  any  factor  common  to 
them ;  and  so  proceed  tilt  the  terms  are  mutually  prime.     Or, 

Divide  both  terms  by  their  greatest  common  divisor. 

2.  Reduce  — ,-'^  to  its  lowest  terms.  Ans.   — 

x-y^  '  X  7f 

6.  Keduce  -r,r^i — - ,  to  its  lowest  terms.    Ans.   -—„ — 
408  or  X  y^  3  a'  y 

24  X  ^  z  2  '^ 

4,  Reduce  ~ — '—  to  its  lowest   terms.  Ans.  — ^ 

Vlaxy  a 

6.  Reduce  — — ^ ,—     to  its  lowest  terms. 
ol  a-b  X  y 


60  ELEMENTARY   ALGEBRA. 

c     -n   J  108  3^  y- 2^ 

0.  Keduce  120 ab~*l?        ^^^  lowest  terms. 

jpf «« 

1.  Reduce  ^  .   »,      ■    a  to  its  lowest  terms. 

Ans.        , 

o-Dj  a»  — 6*  .,  ^  +  y 

o.  Keduce  -^-   _ — i-T-T»  to  its  lowest  terms. 
or  —  2 ao  -f-  tr 

9.  Reduce  — -^ — _^     -^    — ^ to  its  lowest 

terms.  .  a  a:* 

Ans 


X  — .y 


c  (2  x«  +  4  a-) 


10.  Reduce  — -^ — a*—"** to  its  lowest  terms. 


CASE   II. 

86.  To  reduce  fractions  to  equivalent  fractions  having 
a  common  denominator. 

1.    Reduce  r—   and   r-   to  equivalent  fractions  having  a 
common  denominator. 

OPERATION.  We    multiply    the    numerator    and 

Q    a^bjc  denominator  of  each  fraction   by   the 

hy         V^xy  denominator    of   the    other   (Art.    83, 

I  Cor.).      This    must    reduce    them    to 

c    0  c  y  / 

^  -  —  53  ^  equivalent  fractions  having  a  common 

denominator,  as  the  new  denominator 
of  each  fraction  is  the  pro(iuct  of  the  same  factors. 

0^>  In  the  second  operation  we  find  the 

a    ax  least    common    multiple,    bxy^  of  the 

dy        bxy  denominators  by  and  frar;  as  each  de- 

c  cy  nominator   is  contained  in  this'multi- 

J^         j^^  pie,  each  fraction  can   be  reduced  to 

a  fraction  with  this  multiple  as  a  de- 
nominator, by  multiplying  its  numerator  and  denominator  by  the 
quotient  arising  from  dividing  this  multiple  by  its  denominator. 
Hence, 


FRACTIONS.  67 

RULE. 

Multiply  all  the  denominators  together  for  a  common  de- 
nominator, and  multiply  each  numerator  into  the  continued 
product  of  all  the  denominators,  except  its  own,  for  yiew 
numerators.     Or,  . 

Find  the  least  common  multiple  of  the  denominators  for 
the  least  common  denominator.  For  new  numerators,  mul- 
tiply each  numerator  by  the  quotient  arising  from  dividing 
this  multiple  by  its  denominator. 


2.  Reduce        ;    — r»    and     —7—    to    equivalent    fractions 
xy      ab  aoy  ^ 


having  the  least  common  denominator. 

ahm        n : 
abxy     abxy'    """^     abxy 


.  abm        n  X  y  ,         7? 

Ans.     -. —  f    -- ---,    and     — , 


3.  Reduce  -— ,»    --,-»    and     —^ -.    to  equivalent   frac- 

15  0      10  ?>  c  25  ac  a  ^ 

tlons  having  the  least  common  denominator. 

,  80  a'^  erf        Aoadxy  ,        \2hx 

Ans.    -^ — ^,^,»    _„     ,    •  .    and 


IbQabcd     150  abed  150  abed 

4.  Reduce    — >    >    and    --;    to     equivalent     fractions 

m       n  x  y  5  a  ^ 

having  the  least  common  denominator. 

5.  Reduce    -j--  and     to  equivalent  fractions  hav- 
ing the  least  common  denominator. 

.  a^  —  2  ab  -{-b"^         ^  a^m  -\-  abm 

Ans.  7, — ^ and  :r^ — 7^-^- 

a^  —  b^  a-  —  6^ 

6.  .Reduce  ^      and    — ~t_~    to    equivalent    fractions 

X  —  4  X  —  1  ^ 

having  the  least  common  denominator. 

Y.  Reduce    , ;;>   — ; — .    and    to  equivalent   frac- 

ar  —  y     X  -\-  y  x  —  y  ^ 

tions  having  the  least  common  denominator. 


68  KLEAIENTARY    ALGKBKA. 

CASE     III. 

87.   To  add  fractions. 

h  c 

1.  Find  the  sura  of  -    and    -• 

X  X 

OPERATION.  If  anything  is  divided  into  equal 

5  c  h  -\-  c  parts,  a  number  of  these  parts  rep- 

~x  ~^  1:  X  resented  by  h,  added  to  a  number 

represented   by  c,   gives   h  -\-  c  of 

these  parts.     In  the  example  given,  a  unit  is  divided  into  x  equal 

parts,  and  it  is  required  to  find  the  sum  of  h  and  c  of  these  parts ;  i.  e. 

h   ^.     c h  -\-  c 

X   ~^  X  x 

It  is  evident,  therefore,  that  fractions  that  have  a  common  denom- 
inator can  be  added  by  adding  their  numerators.  But  fractions  that 
do  not  have  a  common  denominator  can  be  reduced  to  equivalent 
fractions  having  a  common  denominator.     Hence, 

RULE. 
Reduce  the  fractions,  if  necessary,  to  equivalent  fractions 
having  a  common  denominator;  then   icrite   the  sum   of  the 
numerators  over  the  common  denominator. 

o       Ajj^      ^  J    ^  A  bmy-\-bnx-\-any 

2.  Add    -;    -,    and    ^-  Ans.  '    . — — ' -• 

n      y  h  bny 

3.  Add   -— >    ,^>    and    — -• 

1       Ah  2d 

A       Ajj3a&     2x5/  J        7  m 

4.  Add    .  -  ,    - — •',    and 


Axy     b  ab  S  abx y 

0  g'  b'  - 

40  abx  y 


30a'i-4- 16r»y2_i_35m 
Ans.   —         '  ■        


5.    Add    -    {,   - — J,    and    ^_  — 
3  a*       dcd  27  a c 


6.  Add  - — I —  and   :; — —  Ans.  ; y 

1  -|-  a              1  —  a  1  —  a« 

7.  Add    ,    *"  ^  and   ,  "7°-  Ans.     ,        .  - 

1  —  a            l-f-a  1  — o^ 


FRACTIONS.  69 

8.  Add    ~"t"    and   — ^. 

7  1  -j-  a:^ 

9.  Add    ^  ,,\    and 

10.  Add > >    and   ^^ Ans    1. 

xy  7JZ  xz 

11.  Add    ^r-i-.   and   -P^,' 

sr  —  y  x^  —  7/ 


Ans. 


^  +  y 


7  X 

12.  Add   m  x  and  ^-  • 

18  a 

Note.  —  Consider  w  x  =  —  ,  and  then  proceed  as  before. 

.  18  a  TTZ  a:  -I-  7  X 

Ans.  ' 

7  /v. 

13.  Add  X  -\-  y  and 


18  a 
7  a: 


a-j-^; 

14.    Add  x''  +  2xy  +  if2indi  --^- . 

Ans    ^''-\-^y  —  ^f  —  f-\-  1, 
a:  — y 

CASE    IV. 

88.    To  subtract  one  fraction  from  another. 

c  h 

1.    Subtract    -  from  -• 

X  X 

OPERATION.  If  anything   is    divided   into  x 

I)         c  h  —  c  equal  parts,   a '  number  of  these 

X         X  X  parts  represented  by  c*,  subtracted 

from  a  number  represented  by  &, 

leaves  b  —  c  of  these  parts ;   i.  e. = Hence, 

XX  X 

B  U  L  E . 
Reduce  the  fractions,  if  necessary,  to  equivalent  fractions 
having  a  common  denominator;  then  subtract  the  numerator 
of  the  subtrahend  from  that  of  the  minuend,  and  write  the 
result  over  the  common  denominator. 


70  ELEMExNTARY  ALGEBRA. 

o     a    1  .        .       7a:      „  ab  .  ah  —  \Acx 

2.  Subtract     -7-     from    —  •  Ans.    x 

4  8  c  8  C 

3.  Subtract     =—     from   _ 

7  X  6  ax 

4.  Subtract   ,„    „    from  -7— • 

19x2  19a 

r     d    1..       ,   29  ac  ^  39  a:  .  273  a:"  —  116  acw 

5.  Subtract  -77-,   from Ans.  ^77-5 ^ 

11  2  a 

6.  Subtract  ,  from  -7-  •  Aus.     .- — -. 

1— a  l-|-a  a^  —  1 

7.  Subtract   -'^—^  from      ,-  • 

X  —  1  X  -\-  1 

o     o   u^       ,     ab-\-bc     „  ab  —  be 

8.  Subtract     ,   -    -,c—  from    „  „ — „-»• 

a^x  —  b^x  a^a^  —  i^x^ 

9.  Subtract r  from  — r—r-  Aus. 


a—b  a  -f  6  6=  — a* 

10.  Subtract  ^7 from    ^   *"    •  Ans.     ,  ,    ,- 

X*  —  1  a;-  —  1  ar-f-1 

x^ 7 

11.  Subtract  16  from    ,  -^ 

I  -\-  X 

Note. —  Consider  IG  =  — -,  and  then  proceed  as  before. 

.  ar^— 16  a;— 23 

Ans. 


^ 3 

12.  Subtract  ,   from  xy. 

a  —  b  ^ 

13.  Subtract  x  -\-  b  from     ^   ,    ,    -        Ans. ,    ,    .  ■ 

'  0  4-  4  0-4-4 


CASE    V. 
89.   To  reduce  a  mixed  quantity  to  an  improper  fraction. 

1.  Reduce  x -\-  ^  to  an  improper  fraction. 

OPERATION.  As  eight  eighths  make 

+  a        Bx    ,    a        8ar-|-a  a  unit,   there  will  be  in 

8  8    ^^  8  8  X    units    eight    times    x 

...        .  8x         ,8a:,    a  8a:-l-a       -_ 

eighths ;  1.  e.  x  =  — ;  and    j  -f  -  =    — ^ —  .     Hence, 


FRACTIO^^S.  71 

RULE. 
Multiply  the  integral  part  by  the  denominator  of  the  frac- 
tion;   to  the  product  add   the  numerator  if  the  sign  of  the 
fraction  is  plus,  and  subtract  it  f  the  sign  is  minus,    and 
under  the  I'esuU  write  the  denominator. 

Note.  —  By  a  change  of  the  language,  Examples  12-14  in 
Art.  87,  and  11-13  in  Art.  88,  become  examples  under  this  case. 
Thus,  Example  12,  Art.  87,  might  be  expressed  as  follows:   Reduce 

ma:-f-,cj      to  an  improper  fraction. 

7 

2.  Reduce  x^  +  4 to  an  improper  fraction. 

Ans.      ^  ~ — 

y 

O,     I     X 

3.  Reduce  25  a  —  2b  x  -\-  — T—  to  an  improper  frac- 
tion. 

4.  Reduce  a  —  1  -J-  to  an  improper  fraction. 

.         a'  —  a 
Ans. 


a  +  l 


^  X  11     I     2,*^ 

5.  Reduce  y  -\ "^   '        to  an  improper  fraction. 

6.  Reduce  — ^ {a  -\-  h)  to  an  improper  fraction. 

.          a^  —  ah 
Ans.  7 

0 

x^  J-  \ 
T.    Reduce  x  —  1  — ~-  to  an  improper  fraction. 

Note.  —  It  must  be  remembered  that  the  sign  before  the  dividing 
line  belongs  to  the  fraction  as  a  whole. 

-        x'-\-l        x'—i—x'  —  l        —2  2 

X — 1 ^^ r— =  — j— T'or j— -'     Ans. 

X-\-l  X  -\-  I  x-\-l  x-{-l 

x^  -\-  I 

8.  Reduce  a:  +  1  —    /*       to  an  improper  fraction. 

9 

Ans. 


1  —X 


72  ELEMENTARY    ALGEBRA. 

2  2? a' 

9.  Reduce  x^  —  2ax  -\-  a^  —  — to  an  improper 

fraction. 

Note.  —  According  to  the  same  principle  an  integral  quantity 
can  be  reduced  to  a  fraction  having  any  given  denominator,  by 
multiplying  the  quantity  by  the  proponed  denominator,  and  under  the 
product  writing  the  denominator. 

10.  Reduce  a;  -j-  1   to  a  fraction  whose   denominator  is 

X  —  1.  t,  ^  —  1 

Ans.     - — -  • 
X  —  1 

11.  Reduce  x —  1  to  a  fraction  whose  denominator  is 
a  —  h. 

12.  Reduce   4:  ax  to   a   fraction   whose   denominator  is 

a'  —  z. 

CASE    VI. 
90.   To  reduce  an    improper   fraction   to   an    integral  or 

mixed  quantity. 

^ 4  fl  X    I    5  c^ 

1.  Reduce  _  ' to  an  integral  or  mixed  quan- 
tity. 

OPERATION. 

a' 
X  —  2a)x'^  —  i:  a  X  -\-  b  a^  {x  —  2  a  -\- 


2a 
x^  —  2  ax 

—  2ax  +  5a^ 

—  2ax  +  4a^ 


As  the  value  of  a  fraction  is  the  quotient  arising  from  dividing  the 
numerator  by  the  denominator  (Art.  82),  we  perform  the  indicated 
division.     Hence, 

RULE. 
Divide  the  numerator  by  the  denominator ;  if  there  is  any 
remainder,  place  it  over  the  divisor,  and  annex  the  fraction 
so  formed  with  its  proper  sign  to  the  quotient. 


FRACTIONS.  73 

2.  Reduce   to  an  integral  or  mixed  quantity. 

Ans.  a  —  4  6. 

3.  Reduce ^. -^    to    an   integral  or  mixed 

quantity. 

4.  Reduce  — ^^  to  an  integral  or  mixed  quantity. 

5.  Reduce    — „    — ' —    to  an  integral  or  mixed 

0*2      j      /Tj  nr      I.     /J* 

6.  Reduce    —^ — — ^- —  to  an  integral  or  mixed  quan- 
tity. 

^      -,    ,          8  a  a; — 106a;  —  5cx    ^  •    .  i  •      j 

Y.   Reduce to  an  integral  or  mixed 

£i  X 

quantity. 
8.    R 
^i^^^^^y-       '  Ans.  2a -26--^ 


8.   Reduce ^-—r to  an  integral  or  mixed 

2a  —  26  ° 


a  —  6 

x' if 

9.    Reduce       — -  to  an  integral  or  mixed  quantity. 
X       y 

x^ ?/' 

10.    Reduce  -  to  an  integral  or  mixed  quantity 

X       y 


CASE    VII. 
91.    To  multiply  a  fraction  by  an  integral  quantity. 

1.    Multiply  ^±^  by  c. 

OPERATION.  According  to  the  theorem 

x^y  cx  +  cy  ^"  ^^^'  ^^'  "multiplying  the 

q^     X  g —    ~ab^  numerator   by   c    multiplies 

the  value  of  the   fraction  c 
times. 


74  ELEMENTARY   ALGEBRA. 


2.    Multiply   ~+ ^  by  a. 


OPERATION.  According  to  the  theorem 


ah      ^  ^  —  — h —  nominator   by   a    multiplies 

the  value  of  the  fraction  a 


in  Art.  83,  dividing  the  de- 
nominator  by 
the  value  of  th 
times.     Hence, 


RULE. 
Divide  the  denominator  by  tlie  integral  quantity  when  it 
can  be  done  without  remainder;  otherwise,  multiply  the  nu- 
merator by  the  integral  quantity. 

3.  Multiply  ,1   ,-^  by  m  +  n. 

Ans.   -^- ^• 

m  —  n 

4.  Multiply   ^^i^^by  ai. 

5-   Multiply   3^^  by  %y. 

Note.  —  Any  factor  common  to  the  denominator  and  multiplier 
may  be  cancelled  from  both  before  multiplying. 

-7.    Multiply    ,lJ[^y.^  by  l4(^*-y^). 

Ans.  2(x2  — ^2)  (a  +  ar). 

8.    Multiply    ^-t^  by  x  —  y. 

Note.  —  When  a  fraction  is  multiplied  by  a  quantity  equal  to  its 
denominator,  the  product  is  the  numerator. 


—  X(.r-i,)  =  -+-^  =  x  +  ^,   Ans. 


FRACTIONS.  T5 

9.    Multiply   "^Stlll+Jl  by  (x  -  ay. 

10.    Multiply    "^^  hj  x'  —  2xi/  +  f. 
X       y 

Ans.   {a  -\-  b)  {x  —  y). 

CASE   VIII. 
92.    To  multiply  an  integral  quantity  by  a  fraction. 

1.  Multiply  x2  +  2xy  +  2/'  by  ^-p^- 

OPERATION. 

4(a:2  +  2:ry  +  2/')  -  (^  +  y)  =  ^  (x  +  y) 

We  first  multiply  the  multiplicand  by  the  numerator  4 ;  but  the 
multiplier  is  4  -^  (a;  -[-  ?/)  ;  and  therefore  this  product  is  x  -f-  ^  times 
too  great,  and  this  product  divided  hy  x  -\-  y  must  be  the  product 
sought. 

It  is  evident  that  the  result  would  be  the  same  if  the  division  were 
performed  first,  and  the  multiplication  afterward.     Hence, 

RULE. 
Divide  the  integral  quantity  hy  the  denominator  when  it 
can  he  done  without  remainder,  and  multiply  the  quotient 
hy  the  numerator.  Otherwise,  multiply  the  integral  quantity 
hy  the  numerator,  and  divide  the  product  hy  the  denom- 
inator. 

2.  Multiply  a«  -  3  a^  &  +  3  «  6^  -  ^>^  by  -^,  _  '^^^_^  ^, 

Ans.   7  X  {a  —  h). 

3.  Multiply  a'  —  x'  ^y^^If^- 

3 

4.  Multiply  7  a^  —  4  a?y  by  ^  o_^  • 

5.  Multiply   \n{x^  —  f)   hy^-^l- 


76  ELEMENTARY   ALGEBRA. 

Note. —  Since  the  product  is  the  same,  whichever  quantity  is  con- 
sidered as  tiie  multiplier,  by  considering  the  inte<^ral  quantity  as  the 
multiplier,  Case  VIII.  becomes  the  same  as  Case  VII. 

CASE  IX. 
93i    To  divide  a  fraction  by  an  integral  quantity. 


Divide  -,  by  a. 


OPERATION 

a  1 

6^«  =  6- 


According  to  the  theorem  in  Art. 
83,  dividing  the  numerator  by  a  de- 
creases the  value  of  the  fraction  a 
times. 


Divide   r-  by  c. 


OPERATION. 


a 

re 


According  to  the  theorem  in  Art 
83,  multiplying  the  denominator  by  c 
decreases  the  value  of  the  fraction  c 
times.     Hence, 


RULE. 
Divide  the  numerator  by  the  integral  quantity  when  it  will 
divide  it  without  rernainder ;  otherwise,  multiply  the  denom- 
inator by  the  integral  quantity. 


3.  Divide   ^ -y-    by  a. 

7  X* 

4.  Divide      ,,   by  14^^^- 

6.    Divide  -^ —  by  Q  abc. 

6.    Divide  ^^  by  9  a  6y*. 

62  0  y 

1.   Divide  jf§^5  by  2  («  +  x)  (x  +  y). 

Ans. 


Ans. 

Ans. 


Ans. 


46c 
2/ 


3  J 


32  6»y» 


13(x-|-y)« 


FRACTIONS.  77 

CASE     X. 
94.   To  divide  an  integral  quantity  by  a  fraction. 

1.    Divide  x  by    - 


X    .        ,       ,.  .       . 
a:  -7-  a  =  -  ;  but  the  divisor  is  not  a, 


OPERATION. 

X  but   a  -^b.      Dividing  hs   a,   therefore, 

iC     *      05  -  ....  . 

a  is   dividing    by   a    divisor   &    times    too 

X  hx  great,  and  the  quotient  will  be  h  times 

a^  ~a  *oo  small;  therefore  the  quotient  sought 

.   X       -        hx      ^^ 
IS  -  X  o  =  — •     Hence, 
a  a 


RULE. 
Divide  the  integral  quantity  by  the  numerator,  and  mul- 
tiply the  quotient  by  the  denominator, 

2.  Divide  4: ax  by  — -•  Ans.  — -• 

3.  Divide  7  x'^  by  —f—  •  Ans.  — r—  • 

4.  Divide  a  +  5  by  -  .  Ans.  ""J^tAl, 
.  5.  Divide  a'-\-2ax  -|.  x^  by  "^i^- 

6.  Divide  2:''  —  hx^  by  -• 

7.  Divide  2a:'  +  3/  by  ^^±1. 

8.  Divide  1  by  -.  Ans.  ^. 

Note.  —  Hence,  the  reciprocal  of  a  fraction  is  the  fraction  inverted, 

CASE    XI. 
95.    To  multiply  a  fraction  by  a  fraction. 
a  ,       X 


1.  Multiply  ^  by  -. 


78  ELEMENTARY  ALGEBRA. 


OPERATION.  We  first  multiply  7  by  ar ;  but   the 

a              ax  multiplier  is  not  x,  but  x  ~  y^  there- 

b                       b  fore  the  product  is  y  times  too  great ; 

^  _^  „  ^^  and  -~  —  y  =  -—  (Art.  93)  must  be 


b      '    ^  ~  by 


the  product  sought.     Hence, 


RULE. 


Multiply  the  numerators  together  for  a  new  numerator, 
and  l}ie  denominators  for  a  new  denominator. 

'Note  1.  —  Common  factors  in  the  numerators  and  denominators 
may  be  cancelled  before  multiplication. 

Note  2.  —  Cases  VII.  and  VIII.  can  be  included  in  this  by  writ- 
ing the  integral  quantity  as  the  numerator  of  a  fraction,  with  a  unit 
as  the  denominator. 

Note  3. — Mixed  quantities  may  be  reduced  to  improper  fractions 
before  multiplying. 

2.    Multiply    ,      by   ^  •  Ans. 


be     '^    dy  '    bcdy 

3.  Multiply  ^/,  by  ^^y 

A      HT  ix'  1  rri^ X  ,       ac 

4.  Multiply  —r-  by 

^  "^  abc     ''    mx 

6.    Multiply  J^     by  — - —  Ans.  — ^ — 

6.    Multiply  ^^  by  ^^^. 

^  -^  Axyz     "^    X  —  y 

1.   Multiply  -j-^-+*.-j,;  by  ^  J-j. 

8.  Multiply  ^fby  ^^-. 

9.  Multiply  ^^^  by  ,,,^.     A...  ^^ 


FRACTIONS.  79 

10.  Multiply        ;,,^_4       by  j^^.^,^,^^- 

11.  Multiply   -7^^4-  by  jl,.  Ans.  ^,-^-^. 

12.  Multiply   ^'by^\ 

13.  Multiply  3,  4-  ^4^  by  2,  -  ^. 


Ans.  -Y^ 3^. 

14. 

Multiply  together      ^   ^  ^^ip^'  aad  ^+^. 

Ans.   1, 

15. 

Multiply  together  ''J^^    a^4z7^'  *"^  ^' 

16. 

Multiply  together  a  +  - »  6  +  -  ,  and  y  —  ^  • 

CASE    XII. 
96t   To  divide  a  fraction  by  a  fraction. 

1.    Divide  -  by  -r  • 

y     -^    b 

OPERATION.  r  r 

--4-a=— -  (Art.  93);   but  the 

XX  y  c-y 

-T-  a  •=■  —  ,.  .       .  ,       a  t 

y  ay  divisor  is  not  a,  but  -;  we  nave  used 

b 

_^  sy  1  ^  ^  divisor  b  times  too  great,  and  there- 

"^  ^  fore  the  quotient  —  is  6  times  too 

ay 

X  b  X 

Bmall,  and  the  quotient  sought  is    —  X  &  ==     ^   (Art.   91).     It  will 

'  ^  °  ay  ay  ^  ■' 

be  noticed  that  the  denominator  of  the  dividend  is  multiplied  by  the 
numerator  of  the  divisor,  and  the  numerator  of  the  dividend  by  the 
denominator  of  the  divisor.     Hence, 

RULE. 
Invert  the  divisor,  and  then  proceed  as  in  multiplication  of 
a  fraction  by  a  fraction. 


80  ELEMENTARY   ALGEBRA. 

Note  1. —  All  cases  in  division  of  fractions  can  be  brought  under 
this  rule,  by  writing  integral  quantities  as  fractions  with  a  unit  for 
the  denominator. 

Note  2.  —  After  the  divisor  is  inverted,  common  factors  can  be 
cancelled,  as  in  multiplication  of  fractions. 

Note  3.  —  Mixed  quantities  should  be  reduced  to  improper  frac- 
tions before  division. 

2.  Divide  -   by    -  •  Ans. 

c      "^     n  cm 

ar*  a^  4 

3.  Divide  -  by   t*  -A-ds.  -^~ - 

4.  Divide  -—=—  by 


17  1^       -^      3Ai/ 
5.    Divide    ^^  '. —   by   _  -•  Ads.  — — . .   '      _a  • 


6.   Divide  ^^^3^  by  -^ 


X  — 

■y 

a-  — 

ft» 

4 

X 

mr  —  n" 
n.    Divide   ^  by  ^+^.  Ads.  ^^^ 

8.  Divide   ^-±^  by 

9.  Divide   -rr—x — X  oy  — i Ads.  ^ j — j- 

10.  Divide  ^^^   by  ^^  • 

X  -{-  1        ''         4 

11.  Divide  -^  by   ^7^, 

12.  Divide   -^ r 5-^ — 5  by  — ^ -^^ 

Ads.  3(m«  +  n«). 


13.   Divide   1  +  ^-±^  by*  — ^^ 1. 


A„8.  ^J^4:y)  +  (^  +  y)' 

c(c  —  x  —  y) 


14.    Divide  x  +  y  —  -^  by  ;^  +  a:  -4-  y. 


FRACTIONS.  81 

15.   Dmde   ^-^  +  ^-^  by  j-p^. 

Ads    ^^^+^y 

Note.  —  The  division  of  fractions  is  sometimes  expressed  by  writ> 

a 

ing  the  divisor  under  the  dividend.     Thus,  — .     Such  an  expression 

y 
is  called  a   Complex   Fraction.      A  Complex   Fraction  can  be 

reduced  to  a  simple  one  by  performing  the  division  indicated. 


16.   Reduce  -^  to  a  simple  fraction.  Ans.   =— • 

1                       ^  7  X 
5 

i+? 

It.   Reduce r-  to  a  simple  fraction. 

7 .  cx-\-  ac 

c  Ans. ' — r- . 

i  ex  —  ox 


18. 

Reduce   ^_  ^    to  a  simple  fraction. 

x-\-l 
1 

19. 

Reduce  —  .    .    to  a  simple  fraction. 

1  —X 

a^  —  V" 

20. 

Reduce    ^  ,  '^^    to  a  simple  fraction. 

Ads.  — i 

xJry 

21. 

Reduce  — ^^  to  a  simple  fraction. 
x  —  y                  ^ 

a +  6                           Ans.   {x -\- y)  {a -\- b) . 

]S5"0TE.  —  A  Complex  Fraction  can  also  be  reduced  by  multiplying 
its  numerator  and  denominator  by  the  least  common  multiple  of 
the  denominators  of  the  fractional  parts.  Thus,  if  both  terms  of 
the  fraction  in  Ex.  16  be  multiplied  by  5  a:,  or  both  in  Ex.  17  by 
car,  the  result  will  be  the  same  as  above. 

4*  F 


82  ELEMENTARY   ALGEBRA- 


SECTION    XIII. 

EQUATIONS 

OF  THE  FIRST  DEGREE  CONTAINING  BUT  ONE  UNKNOWN 
QUANTITY. 

97.  An  Equation  is  an  expression  of  equality  between 
two  quantities  (Art.  9).  That  portion  of  the  equation 
which  precedes  the  sign  =  is  called  the  first  member,  and 
that  which  follows,  the  second  member. 

98.  The  Degree  of  an  equation  containing  but  one  un- 
known quantity  is  denoted  by  the  exponent  of  the  highest 
power  of  the  unknown  quantity  in  the  equation. 

An  equation  of  the  first  degree,  or  a  simple  equation,  is 
one  that  contains  only  the  first  power  of  the  unknown 
quantity.     For  example, 

An  equation  of  the  second  degree,  or  a  quadratic  equation, 
is  one  in  which  the  highest  power  of  the  unknown  quantity 
is  the  second  power.     For  example, 

x^  —  aa;  =  i-|-c,  or  ax^  —  Jr=17. 

An  equation  of  the  third  degree,  or  a  cubic  equation,  is  one 
in  which  the  highest  power  of  the  unknown  quantity  is  the 
third  power,  and  so  on. 

99.  The  Reduction  op  an  Equation  consists  in  finding  the 
value  of  the  unknown  quantity,  and  the  processes  involved 
depend  upon  the  Axioms  given  in  Art.  13.  The  processes 
can  be  best  understood  by  considering  an  equation  as  a  pair 
of  scales  which  balance  as  long  as  an  equal  weight  remains 
in  both  sides  :  whenever  on  one  side  any  additional  weight 
is  put  in  or  taken  out,  an  equal  weight  must  be  put  in  or 


EQUATIONS   OF   THE   FIRST   DEGREE.  83 

taken  out  on  the  other  side,  in  order  that  the  equilibrium 
may  remain.  So,  in  an  equation,  whatever  is  done  to  one 
side  must  be  done  to  the  other,  in  order  that  the  equality  may 
remain. 

1.  If  anythinp^  is  added  to  one  member,  an  equal  quantity 
must  be  added  to  the  other. 

2.  If  anj^thing  is  subtracted  from  one  member,  an  equal 
quantity  must  be  subtracted  from  the  other. 

3.  If  one  member  is  multiplied  by  any  quantity,  the  other 
member  must  be  multiplied  by  an  equal  quantity. 

4.  If  one  member  is  divided  by  any  quantity,  the  other 
member  must  be  divided  by  an  equal  quantity. 

5.  If  one  member  is  involved  or  evolved,  the  other  must 
be  involved  or  evolved  to  the  same  degree. 

TRANSPOSITION. 

lOOo  Transposition  is  the  changing  of  terms  from  one 
member  of  an  equation  to  the  other,  without  destroying 
the  equality. 

The  object  of  transposition  is  to  bring  all  the  unknow^n 
terms  into  one  member  and  all  the  known  into  the  other, 
so  that  the  unknown  may  become  known. 

1,  Find  the  value  of  x  in  the  equation  x  -\-  16  =  24, 

Subtracting  16  from  the  first 

OPERATION.  member  leaves  x ;  but  if  1 6  is  sub- 

a:  -f-  16  =  24  tracted  from  the  first  member,  it 

ar  =  24  —  16  =  8       must  also  be  subtracted  from  the 

second. 

2.  Find  the  value  of  x  in  the  equation  x  —  h^=a. 

OPERATION.  Adding  h  to  the  first  member 

^ J  =  a  gives  x  ;  but  if  h  is  added  to  the 

^  1     T  first  member  it  must  also  be  added 

X  =  a  -\-  0 

to  the  second. 


84  ELEMENTARY  ALGEBRA. 

3.  Find  the  value  of  x  in  the  equation  2  x  ==  x -\-  16. 

OPERATION. 

22;  z=  X  4-  16  Subtracting  x  from  both  mem- 

2x xzz^ia  bers,  we  have  2ar  — x=  16,  or 

i«  a:  =16. 

cc  =  16 

It  appears  from  these  exan)ples  that  any  term  which  dis- 
appears from  one  member  of  an  equation  reappears  in  the 
other  with  the  opposite  sign.     Hence, 

RULE. 

Amj  term  may  be  transposed  from  one  member  of  an 
equation  to  the  other,  provided  its  sign  is  changed. 

4.  Find  the  value  of  x  in  the  equation  8a:  —  15  =  4a;  +  o. 

OPERATION. 

8a;  — 15  =    4a:-f    5 
Transposing,  8a: — 4a:=r    5     +15 

Uniting  terms,  4x=20 

Dividing  both  members  by  4,  a:=    5 

5.  Find  the  value  of  a:  in  4  a:  -f-  46  =  5  a:  -|-  23. 

Note.  —  Reducing,  we  liave  —  x  =  —  23.  If  each  member  of 
this  equation  is  transposed,  we  shall  have  23  =  a: ;  i.  e.  23  equals 
X,  or  X  equals  23.  Dividing  both  members  by  —  1  will  give  the 
same  result.  Hence,  the  sic/ns  of  all  the  terms  of  an  equation  may 
be  changed  without  destroying  the  equality. 

6.  Find  the  value  of  x  in  17  a;  +  IT  =  19  or  +  13. 

Alls.  x=z2. 
1.    Find  the  value  of  a;  in  8  a:  —  14  =  13  a:  —  29. 

8.  Find  the  value  of  a:  in  5  a:  +  25  =  10  ar  —  25. 

Ans.  X  =  10. 

9,  Find  the  value  of  .r  in  24a;  —  IT  =  11  a:  +  74. 

10.    Find  the  value  of  x  in  3T  a:  —  (4  +  T)  =  41  a:  —  23. 


EQUATIONS   OF  THE  FIRST  DEGREE.  85 

CLEARING   OF   FRACTIONS. 
101  •    To  clear  an  equation  of  fractions. 

1.  Find  the  value  of  x  in  the  equation  -  —  2  =  --J-  1. 

o  o 

OPERATION.  If  the  given  equation  is  mul- 

^ 2  =  -  4-  1  tiplied  by  6,  the  least   common 

^  ^  multiple  of  6  and  3,  it  will  give 

2x  —  12  =zx  -{-  6  2x  —  12  =  a:-|-6,  an  equation 

a;  1=  18  without  a  fractional  term.  Hence, 

RULE. 

Multiply  each  term  of  the  equation  by  the  least  common 
multiple  of  the  denominators. 

Note  1.  —  In  multiplying  a  fractional  term,  divide  the  multiplier 
by  the  denominator  of  the  fraction  and  multiply  the  numerator  by 
the  quotient. 

Note  2  —  An  equation  may  be  cleared  of  fractions  by  multiplying 
it  first  by  one  denominator,  and  the  resulting  equation  by  another,  and 
so  on,  till  all  the  denominators  disappear ;  but  multiplying  by  the 
least  common  multiple  is  generally  the  more   expeditious  method. 

Note  3.  —  Before  clearing  of  fractions  it  is  better  to  unite  terms 
which   can   readily  be  united;    for  instance,  the   equation  in  Ex.  1, 

by  transposing  —  2,  can  be  written  ^  =  ^  -|-  3. 

o         o 
Note  4.  —  When   the    sign  —  is    before  a    fraction  and    the    de- 
nominator  is  removed,  the  sign  of   each   term  that  was  in  the  nu- 
merator must  be  changed. 

2.  Given  ^  —  ^  +  25  =  33  —  ^^  • 


# 

OPERATION. 

Transposing  25, 

X 

4 

X 

5  ~ 

=  8  — - 

—  6 

2 

Multiplying  by  20, 

bx- 

-4:r  = 

=  160- 

-10a; 

+ 

60 

Transposing  and  uniting, 

11.2r  = 

=  220 

Dividing  by  11, 

aj  = 

=  20 

86  ELEMENTARY   ALGEBRA. 

X 

Note.  —  The  sign  of  the  numerator  of  —  r  is  -f"*  ^"^  '""st  be 

o 

changed   to  —  when  the  denominator   is   removed ;    for  —  (-j-  4  x) 

=  —  A  x\  and  so  the  sign  of  each  term  of  the  numerator  of  the  fraction 

X  —  6 
—  — - —   must  be  changed  when  the  denominator  2  is  removed;  for 

—  (-j-  10  a;  —  60)  =  —  10  a:  -[-  60. 

102i  To  reduce  an  equation  of  the  first  degree  contain- 
ing but  one  unknown  quantity,  we  deduce  from  the  preced- 
ing examples  the  following 

RULE. 

Clear  the  equation  of  fractions,  if  necessary. 

Transpose  the  known  terms  to  one  member  and  the  un- 
known to  the  other,  and  reduce  each  member  to  its  simplest 
form. 

Divide  both  members  by  the  coefficient  of  tJie  unknown 
quantity. 

Note  1.  —  To  verify  an  equation,  we  have  only  to  substitute  in 
the  equation  the  value  of  the  unknown  quantity  found  by  reducing 
the  equation.     For  instance,  in  Ex.  2,  Art.  101,  by  substituting  20 


for  X, 

.       X 

m-- 

1  + 

25 

=  33 

X 

—  6 

— — ,  we  have 

20 
4 

■f  + 

25 

=  33 

20 

—  6 

2       ' 

5  - 

-4  + 

25 
26 

=  33 
=  26. 

—  7, 

Note  2.- 
ed. 

-  When 

answers  are  ffbt  | 

given, 

the  work  shoul 

103.  Since  the  relations  between  quantities  in  Algebra 
are  often  expressed  in  the  form  of  a  proportion,  we  intro- 
duce here  the  necessary  definitions. 


'   EQUATIONS    OF   THE   FIRST   DEGREE.  87 

104.  Ratio  is  the  relation  of  one  quantity  to  another  of 
the  same  kind  ;  or,  it  is  the  quotient  which  arises  from  di- 
viding one  quantity  by  another  of  the  same  kind. 

Ratio  is  indicated  by  writing  the  two  quantities  after 
one  another  with  two  dots  between,  or  by  expressing  the 
division  in  the  form  of  a  fraction.  Thus,  the  ratio  of  a  to 
h  is  written,  a  :  b,  or  -  ;  read,  a  is  to  b,  or  a  divided  by  b. 

105.  Proportion  is  an  equality  of  ratios.  Four  quan- 
tities are  proportional  when  the  ratio  of  the  first  to  the 
second  is  equal  to  the  ratio  of  the  third  to  the  fourth. 

The  equality  of  two  ratios  is  indicated  by  the  sign 
of  equality  (=)  or  by  four  dots  (::). 

(1         c 

Thus,  a  :  b  ^=  c  :  d,  ov  a  :  b:  :  c  :  c?,  or  y  =  -^ ;  read,  a  to  6 
equals  c  to  d,  ,or  a  is  to  6  as  c  is  to  d,  ova  divided  by  b 
equals  c  divided  by  d. 

The  first  and  fourth  terms  of  a  proportion  are  called  the 
extremes,  and  the  second  and  third  the  means. 

106.  In  a  proportion  the  product  of  the  means  is  equal 
to  the  product  of  the  extremes. 

Let  a  :  b  z=  c  :  d 

i.  e. 

Clearing  of  fractions, 

A  proportion  is  an  equation  ;  and  making  the  product 
of  the  means  equal  to  the  product  of  the  extremes  is 
merely  clearing  the  equation  of  fractions. 

Examples. 
.    1.    Reduce  ^  +  10  =  I-^  +  13.  Ans.  rr  =  30, 

2.   Reduce  17  a;  —  14  =  12  cc  —  4.  Ans.  x  =  2. 


a 
b 

c 
d 

ad 

be 

88  ELEMENTARY    ALGEBRA. 

3.  Reduce  6  a:  —  25  +  x  =  135  —  3  a:  —  10. 

Ans.  X  =^  15. 

4.  Reduce  3a:  +  5  —  a:  =  38  —  2a:.      Ans.  x  —  8f 

5.  Reduce    ~~-  +  |  =  30  —  "^-i^-    Ans.  x  =  12. 

31         3  X 

6.  Reduce  x  —  Tt]-  = •  Ans.  x  =  IlyV- 

7.  Reduce  f  +  ^  +  '^  +  ^=  154.         Ans.  x  =  120. 

2         3         4         5 

8.  Reduce   ^  +  ^  =  16  +  ^ .  Ans.  x  =  24. 

9.  Reduce  --l-«=r-  —  '  -4-  d. 

OPERATTON. 

Multiplying  by5c^,     cAx-|-«ic^==:icx  —  b  hx-\-bc  dh 
Transposing,       c  kx  —  b  c  x-\-hhx=ibc  d  h  —  abch 
Factoring  1st  mem.,  (c  A  —  bc-\-bh)x=zbc  d  h  —  abch 

b  c  dli  —  abch 


Dividing  by  coeflScient  of  x,  x  = 


c  h  —  b  c  -\-b  h 


10.  Reduce  x-\-mx  =  c.  Ans.  x  = —-, — . 

'  1  -f-  ^» 

11.  Reduce 3  =  7.  Ans.  x  =    ~    » 

X  10 

12.  Reduce  — \--z=x.  Ans.  x=   -     •"    . 

a       c  a  c 

13.  Reduce      4- -  r=  c.  Ans.  a:  =  ^  *"    • 

X     ^     X  c 

14.  Reduce  8  =  -^—-  +  6.  Ans.  x  =  9. 

x—  2    "^ 

ieT.j           23a  .                     «4-l 

15.  Reduce = c.  Ans.  x  =  — ' —  • 

XXX  c 


EQUATIONS   OF   THE   FIRST   DEGREE.  89 

16.  Reduce  | -f  |  +  |  =  39. 

17.  Reduce  — --l-c  =  d. 

a         0    ' 

X         1 

18.  Reduce  {a  —  &)  x  -| —  =  -  • 

19.  Reduce  ar  —  /|  —  |)  =  6.  Ans.  a:  =  6. 

20.  Reduce  6  —  ^^^  =  x  —  4. 

5 

oi      T?   J  o  Ox  — 29         _Q         6a: -1-11 

21.  Reduce  2  x ^ ==  18 i 

4  . 5 

Ans.  a;  =  9. 

■    ^     _,    -        •  117— X         „  a:— 95 

22.  Reduce  =  3  a?  -[-         — 

23.  Reduce  2  a: ^^  —  14 ^"p—     ^"s.  x=zb. 

24.  Reduce  6x-hn-|  =  9^-T+^- 

Note.  —  Before  clearing  of  fractions,  transpose   7^  and  unite  it 

.,,,  X        .....llx 

with    9^;   also   transpose  —  -,  and  unite  it  with    — -. 

25.  Reduce  4  a:  -j ' —  =  5  -| ^- 

26.  Reduce  ^^  —  ^^  =  21  —  ^i?.    Ans.  x  =  39. 

2  6  6 

27.  Reduce  — [-  -:  -\-  -  =  d.      Ans.  a:  =  r — ; ; — v 

a    *    o    ^    c  be  -^  ac-\-  ab 

28.  Reduce  — ~-  —  6  =  ^^  4-  Y. 

3  4  ' 

onT^j  a;— 1  _        22  —  a:         34-a:         .  _ 

29.  Reduce  — - —  =  6 ^ i —     Ans.  a:  =  7. 

6  5  5 


90  ELEMENTARY   ALGEBRA. 

on      r>   ^  in     I     2a;— 22         3x—75    ,     284  — 4 a: 

30.  Reduce  19  -| —  = 1 ~ . 

oi      -D    J  4a:  4-5         5a:  —  5         x  4- 1 

31.  Reduce — s^ —  =  — ~ 1. 

5  4  o 

Ans.  X  =  5. 

««      T^    1  18  — 5a:         3x4-3     *  ,^     ,    5x4-3 

32.  Reduce ^ ^  =  4a:  —  17  -\ ^• 

««     T.    J  .  a:— 12    ,     ^  20x4-21         1 

33.  Reduce  4  a; h  ^  ==  ?- t  • 

o  '  4  4 

34.  Reduce =  -  •  Ans.  x  = 


X        c        m  '  bm  -\-  cd 

X  CL  X 

35.  Reduce  v ^^  :=  1  —  3  a  c. 

h         c 

oa      ^   A  5x'-|-3     ,     _         3X-I-15  6x4-10 

36.  Reduce  — ^^—  -f-  6 ^ =  4  +  — ^—  • 

OK      r»    1  o  3x— 19        „  23— X    ,    5x  — 38    ,    ,^ 

37.  Reduce  3  a; ^r 8  = ^- 10. 

J  4  3 

Ans.  X  =  19. 

«o      T.   J  13  — 3x        3x4-2         ^         ^        ,    8x— 13 

38.  Reduce        ,„ J— =  7  —  6  x  4 

«^      T,    J  4  (x  —  7)    ,    3  (x  4-  1)         7  X  —  1 7    ,     X 

39.  Reduce    -^-^  +    -^j"^^  =  — ^y-  +  21  * 

^^      _,    ,  4x  — 6    ,    _         19— 4x        5x  — 6    ,    7x-|-8 

40.  Reduce  a: \-^  =  — ^3 j 1 ^ — 

41.  Reduce  -+tH [-->  =  '». 

a    *    b    *    c    '    d 

.0      r»    ^  7X-I-5    ,     6x— 30  .     , 

42.  Reduce    -  ^' 1-   ^  ^  __^    =  a:  +  1. 

Note.  —  Multiply  by  7,  transpose,  and  unite. 

43.  Reduce  2  (3  -f  a:)  :  6  a:  —  9  =  2  :  3.  Ans.  x=  6. 

44.  Reduce  I  +  I  :  ,-^+4^  =  11  :  I- 

45.  Reduce b  :  c  4-  d  =      in. 


EQUATIONS   OF   THE   FIRST   DEGREE.  91 


PROBLEMS 

PRODUCING    EQUATIONS    OF    THE    FIRST    DEGREE    CON- 
TAINING   BUT    ONE    UNKNOWN    QUANTITY. 

107i  The  problems  given  in  this  Section  must  either  con- 
tain but  one  unknown  quantity,  or  the  unknown  quanti- 
ties must  be  so  related  to  one  another  that  if  one  be- 
comes known  the  others  also  become  known. 

108i  With  beginners  the  chief  difficulty  in  solving  a 
problem  is  in  translating  the  statements  or  conditions 
of  the  problem  from  common  to  algebraic  language  ;  i.  e. 
in  preparing  the  data,  and  forming  an  equation  in  accord- 
ance with  the  given  conditions. 

1.  If  three  times  a  certain  number  is  added  to  one  half 
and  one  third  of  itself,  the  sum  is  115.  What  is  the 
number  ? 

SOLUTION. 

Let  X  z=  the  number ; 

then  by  the  conditions  of  the  problem, 

3^+1  +  1  =  115 

Clearing  of  fractions,  lSx-\-3x-\-2x=.  690 
Uniting  terms,  23  x  =  690 

Dividing  by  23,  a:  =    30 

VERIFICATI(5N. 

3X  30  +  ^  +  ^^=115 

115  =  115 

In  this  problem  there  is  but  one  unknown  quantity,  which  we  rep- 
resent by  X. 

2.  There  are  three  numbers  of  which  the  first  is  6  more 
than  the  second,  and  11  less  than  the  third  ;  and  their  sum 
is  101.     What  are  the  numbers? 


92  ELEMENTARY   ALGEBRA. 

SOLUTION. 

Let  X  ■^=  the  first,  In  this   problem 

then  X  —    6  =  the  second,  there  are  three  un- 

and  y +  11  =  the  third.  known     quantities  ; 

mi    •  ^i       i      c         if^V  but  they  are  so  re- 

Their  sum,  3  X -4-    6  =  101  ,      ,    -^ 

-  _  lated    to    one    an- 

3  a;  =    96  *u       *u  *     r 

other    that,   if  any 

x=    32.  the  first,  one  becomes  known, 

X—    6  =    26,  the  second,       the  other  two  will 
a:  +  1 1  =    43,  the  third.  be  known. 

VERIFICATION. 

32  +  26  +  43  =  101 
101  =  101 

From  these  examples  we  deduce  the  following 

GENERAL    RULE. 

Let  X  {or  some  one  of  the  latter  letters  of  the  alphabet) 
represent  the  unknown  quantity ;  or,  if  there  is  more  than  one 
unknown  quantity,  let  x  represent  one,  and  find  the  others  by 
expressing  in  algebraic  form  their  given  relations  to  the  one 
represented  by  x. 

With  the  data  thus  prepared  form  an  equation  in  accord- 
ance with  the  conditions  given  in  the  problem. 

Solve  the  equation. 

The  three  steps  may  be  briefly  expressed  thus :  — 
Ist.  Preparing  the  Data  ; 
2d.    Forming  the  Equation  ; 
3d.   Solving  the  Equation. 

3.  The  sum  of  three  numbers  is  960  ;  the  first  is  one 
half  of  the  second  and  one  third  of  the  third.  What  are 
the  numbers  ?  Ans.   160,  320,  and  480. 

4.  Find  two  numbers  whose  difference  is  18  and  whose 
Bum  112.  Ans.   47  and  65. 


EQUATIONS   OF   THE   FIRST    DEGREE.  93 

6.  A  man  being  asked  how  much  he  gave  for  his  horse 
said,  that  if  he  had  given  $  70  more  than  three  times  as 
much  as  it  cost,  he  would  have  given  $445.  How  much 
did  his  horse  cost  him  ? 

6.  A  man  being  asked  how  many  sheep  he  had,  replied 
that  if  he  had  as  many  more,  and  two  thirds  as  many, 
and  three  fifths  as  many,  he  should  have  8  more  than 
three  times  as  many  as  he  had.     How  many  sheep  had  he  ? 

7.  Divide  $575  between  A  and  B  in  such  a  manner 
that  B  may  have  two  thirds  as  much  as  A. 

Ans,  A's  share,  S  345  ; 
B's      "      $230. 

8.  A  father  divided  his  estate  among  his  three  children 
so  that  the  eldest  had  $  1440  less  than  one  half  of  the 
whole,  the  second  $500  more  than  one  third  of  the 
whole,  and  the  youngest  $250  more  than  one  fourth 
of  the  whole.     What  was  the  value  of  the  estate  ? 


SOLUTION. 

Let 

X  =  whole  estate. 

Then 

X 

2  "~ 

1440  =  share  of  the  eldest, 

1  + 

500=     "       "     "     second, 

2  + 

250  =     "       "     "     youngest, 

Their  sum 

13x 
12 

690  =  X,  whole  estate. 
X  =  8280,  whole  estate. 

9.  A  gentleman  meeting  five  poor  persons,  distributed 
$7.60,  giving  to  the  second  twice,  to  the  third  three  times, 
to  the  fourth  four  times,  and  to  the  fifth  five  times  as  much 
as  to  the  first.     How  much  did  he  give  to  each  ? 


94  ELEMENTARY   ALGEBRA. 

10.  Divide  795  into  two  such  parts  that  the  greater  di- 
vided by  3  shall  be  equal  to  the  less  divided  by  2. 

Note.  —  To  avoid  fractions,  let  3  a: = the  greater  and  2x=the  less. 

Ans.  477  and  318. 

11.  Divide  a  into  two  such  parts  that  the  greater  di- 
vided by  b  shall  be  equal  to  the  less  divided  by  c, 

SOLUTION. 

Let  X  :=  the  greater, 

then  a  —  x  =  the  less. 

.     -  X        a  —  X 

And  T  = 

b  c 

Clearing  of  fractions,        c  x  ■=  ab  —  bx 
Transposing,  bx  -{-  c x  :=z  ab 

Dividing  by  b -\- c,  x  =          ■,    the  greater, 

ab  ac      ^ ,       , 

a  —  x^=.  a  —  ,   ,-  =  ,— r^-,  the  less. 
b-\-c         b-\-c 

12.  What  number  is  that  which,  if  multiplied  by  7,  and 
the  product  increased  by  eleven  times  the  number,  and 
this  sum  divided  by  9,  will  give  the  quotient  6  ? 

13.  If  to  a  certain  number  55  is  added,  and  the  sum 
divided  by  9,  the  quotient  will  be  5  less  than  one  fifth 
of  the  number.     What  is  the  number?  Ans.  125. 

14.  As  A  and  B  are  talking  of  their  ages,  A  says  to  B, 
"If  one  third,  one  fourth,  and  seven  twelfths  of  my  age 
are  added  to  my  age,  the  sura  will  be  8  more  than  twice 
my  age/'     What  was  A's  age  ? 

15.  A  farmer  having  bought  a  horse  kept  him  six  weeks 
at  an  expense  of  $20,  and  then  sold  him  for  four  fifths  of 
the  original  cost,  losing  thereby  %  50.  IIow  much  did  he 
pay  for  the  horse?  Ans.  S150. 

16.  A  man  left  $  18204,  to  be  divided  among  his  widow, 
three  sons,  and  two  daughters,  in  such  a  manner  that  the 
widow  should  have  twice  as  mtich  as  a  son,  and  each  son 
as  much  as  both  daughters.     What  was  the  share  of  each  ? 


EQUATIONS    OF   THE    FIRST   DEGREE.  95 

n.  If  a  certain  number  is  divided  by  9,  tiie  sum  of  the 
divisor,  dividend,  and  quotient  will  be  89.  What  is  the 
number?  Ans.   72. 

18.  If  a  certain  quantity  is  divided  by  a,  the  sum  of  the 
divisor,  dividend,  and  quotient  will  be  h.  What  is  the 
quantity?  ^^^    a  h  -  a\ 

•     a-fl 

19.  Verify  the  answer  to  the  preceding  problem. 

20.  A  farmer  mixed  together  corn,  barley,  and  oats.  In 
all  there  were  80  bushels,  and  the  mixture  contained  two 
thirds  as  much  corn  as  barley  and  one  fifth  as  much  bar- 
ley as  oats.     How  many  bushels  of  each  were  there  ? 

21.  Three  men,  A,  B,  and  C,  built  572  rods  of  fence.  A 
built  8  rods  per  day,  B  7,  and  C  5.  A  worked  one  half 
as  many  days  as  B,  and  B  one  third  as  many  as  C.  How 
many  days  did  each  work  ? 

22.  What  number  is  as  much  greater  than  340  as  its 
third  part  is  greater  than  34  ?  Ans.  459. 

23.  A  man  meeting  some  beggars  gave  3  cents  to  each, 
and  had  4  cents  left.  If  he  had  undertaken  to  give  5  cents 
to  each,  he  would  have  needed  6  cents  to  complete  the  dis- 
tribution. How  many  beggars  were  there,  and  how  much 
money  did  he  have  ? 

SOLUTION. 

Let  X  =z  the  number  of  beggars  ; 

then,  according  to  the  first  statement, 

3  a:  -|-  4  =  the  number  of  cents  he  had, 
and,  according  to  the  second  statement, 

^b  X  —  6  =  the  number  of  cents  he  had. 
Therefore,  5a?  —  6  =  3a;  +  4 

2^=10 
X  =z    b,  the  number  of  beggars, 
and  3ar  -|-  4  =  19,  the  number  of  cents  he  had. 


96  ELEMENTARY    ALGEBRA. 

24.  A  boy  wishing  to  distHbute  all  his  money  among 
his  companions  gave  to  each  2  cents,  and  had  3  cents 
left  ;  therefore,  collecting  it  again,  he  began  to  give  3 
cents  to  each,  but  found  that  in  this  case  there  was  one 
who  had  received  none,  and  another  who  had  only  2 
cents.  How  many  companions,  and  how  much  money 
had   he  ?  Ans.    7  companions,  and  17  cents. 

25.  What  two  numbers  whose  difference  is  35  are  to 
each  other  as  4  :  5  ? 

26.  A  man  being  asked  the  hour,  answered  that  three 
times  the  number  of  hours  before  noon  was  equal  to  three 
fifths  of  the  number  since  midnight.  What  was  the  time 
of  day  ? 

SOLUTION 

Let  X  =  the  number  of  hours  since  midnight,  i.  e.  the  time  ; 
then  12  —  x  =z  the  number  of  hours  before  noon. 

Then  36—    Sx=^-^ 

o 

Clearing  of  fractions,    180  —  15  a:  =  3  a: 

Whence  18  a:  =180 

X  =    10.     Ans,   10  o'clock. 

27.  A  gains  in  trade  $300;  B  gains  one  half  as  much 
as  A,  plus  one  third  as  much  as  C  ;  and  C  gains  as  much 
as  A  and  B.     What  is  the  gain  of  B  and  C  ? 

Ans.  B's,  $375;  C's,  $675. 

28.  What  number  is  to  28  increased  by  one  third  of 
the  number    as  2  :  3  ?  Ans.  24. 

29.  What  number  is  that  whose  fifth  part  exceeds  its 
sixth  by  15? 

30.  Divide  $3740  into  two  parts  which  shall  be  in  the 
ratio  of  10:7. 

31.  Divide  a  into  two  parts  which  shall  be  in  the  ratio 

of  b  :  a  .  ab  .,     ac 

Ans.  ,— . —  and 


bJ^c  b-irc 


EQUATIONS   OF   THE   FIRST   DEGREE.  97 

32.  What  number  is  that  the  sum  of  whose  fourth  part, 
fifth  part,  and  sixth  part  is  37  ? 

33.  What  quantity  is  that  the  sum  of  whose  third  part, 
fifth  part,  and  seventh  part  is  «  ?  ^^^    105  a 

34.  A  farmer  sold  IT  bushels  of  oats  at  a  certain  price, 
and  afterward  12  bushels  at  the  same  rate  ;  the  second 
time  he  received  55  shillings  less  than  the  first.  What 
was  the  price  per  bushel  ? 

35.  A  certain  number  consists  of  two  figures  whose 
sum  is  9  ;  and  if  2t  is  added  to  the  number,  the  order  of 
the  figures  will  be  inverted.     What  is  the  number? 

SOLUTION. 

Let  X  =  the  left-hand  figure  ; 

then  9  —  x  =  the  right-hand  figure. 

As  figures  increase  from  right  to  left  in  a  tenfold  ratio, 
10  a:  +  (9  —  x)  rz=z  9  X  -\-  9  =  the  number  ; 
and  when  the  order  of  the  figures  is  inverted, 
10  (9  —  x)  -\-  X  =z  90  —  9  X  =  the  resulting  number. 
Therefore       9x  +  9  +  2t  =  90  —  9a; 
Or  18  a;  =  64 

Whence  .  a:  =    3,  the  left-hand  figure, 

and  9  —  x=    6,  the  right-hand  figure. 

Ans.  36. 

36.  A  certain  number  consists  of  three  figures  whose 
sum  is  6,  and  the  middle  figure  is  double  the  left-hand 
figure ;  and  if  198  is  added  to  the  number,  the  order  of 
the  figures  will  be  inverted.     What  is  the  number  ? 

Ans.  123. 

37.  Two  men  90  miles  apart  travel  towards  each  other 
till  they  meet.  The  first  travels  5  miles  an  hour  and  the 
second  4.  How  many  miles  does  each  travel  before  they 
meet  ? 


98  ELEMENTARY   ALGEBRA. 

38.  A  man  hired  six  laborers,  to  the  first  of  whom  he 
paid  75  cents  a  week  more  than  to  the  second ;  to  the 
second,  80  cents  more  than  to  the  third  ;  to  the  third,  60 
cents  more  than  to  the  fourth  ;  to  the  fourth,  50  cents 
more  than  to  the  fifth  ;  to  the  fifth,  40  cents  more  than 
to  the  sixth;  and  to  all  he  paid  $68.15  a  week.  What 
did  he  pay  to  each  a  week  ? 

39.  What  number  is  that  to  which  if  20  is  added  two 
thirds  of  the  sum  will  be  80  ? 

40.  What  number  is  that  to  which  if  a  is  added   -    of 

c 

the  sum  will  be  rf?  .         cd 

Ans.   ~ a. 

b 

41.  A  man  spent  one  fourth  of  his  life  in  Ireland,  one 
fifth  in  England,  and  the  rest,  which  was  33  years,  in 
the  United  States.     To  what  age  did  he  live? 

42.  A  post  is  one  fifth  in  the  mud,  two  sevenths  in 
the  water,  and  18  feet  above  the  water.  How  long  is 
the  post  ? 

43.  What  number  is  that  whose  half  is  as  much  less 
than  40  as  three  times  the  number  is  greater  than  156? 

Ans.  56. 

44.  Two  workmen  received  the  same  sum  for  their  la- 
bor ;  but  if  one  had  received  $  15  less  and  the  other  $  15 
more,  one  would  have  received  just  four  times  as  much 
as  the  other.     What  did  each  receive  ? 

45.  Of  the  trees  on  a  certain  lot  of  land  five  sevenths 
are  oak,  one  fifth  are  chestnut,  and  there  are  32  less  wal- 
nut trees  than  chestnut.     How  many  trees  are  there  ? 

46.  Divide  474  into  two  parts  such  that,  if  the  greater 
part  is  divided  by  7  and  the  less  by  3,  the  first  quo- 
tient shall  be  greater  than  the  second  hy  12. 

Ans.  357  and  117. 


EQUATIONS    OF   THE   FIRST   DEGREE.  99 

47.  Two  persons,  A  and  B,  have  each  an  annual  income 
of  S  1500.  A  spends  every  year  $400  more  than  B,  and 
at  the  end  of  five  years  the  amount  of  their  savings  is 
$6000.     What  does  each  spend  annually? 

Ans.  A  $1100,  and  B  $T00. 

48.  In  a  skirmish  the  number  of  men  captured  was  41 
more,  and  the  number  killed  26  less  than  the  number 
wounded  ;  45  men  ran  away ;  and  the  whole  number  en- 
gaged was  four  times  the  number  wounded.  How  many 
men  belonged  to  the  skirmishing  party  ?  Ans.  240. 

49.  A  and  B  have  the  same  salary.  A  runs  into  debt 
every  year  a  sum  equal  to  one  sixth  of  his  salary,  while 
B  spends  only  three  fourths  of  his  ;  at  the  end  of  five 
years  B  has  saved  $  1000  more  than  enough  to  pay  A's 
debt.     What  is  the  salary  of  each  ?  Ans.  $  2400. 

50.  A  man  lived  single  one  third  of  his  life:  after  hav- 
ing been  married  two  years  more  than  one  eighth  of  his 
life,  he  had  a  daughter  who  died  ten  years  after  him, 
and  whose  age  at  her  death  was  one  year  less  than  two 
thirds  the  age  of  her  father  at  his  death.  AVhat  was  the 
father's  age  at  his  death  ? 

SOLUTION. 

Let  X  =  his  age  j 

then  -  =r  his  age  at  marriage, 

o 

-  -f-  -  -|-  2  =  his  age  at  daughter's  birth, 
and  X  —  f  ^  -)~  u  +  2 )  =  her  age  at  his  death. 

Then         ._f_^_2+10=.L--l 
Transposing  and  uniting,  —  -  =  —  9 

o 

X  =  72,  the  father's  age. 


100  ELEMENTARY   ALGEBRA. 

51.  Divide  S  864  among  three  persons  so  that  A  shall  have 
as  much  as  B  and  C  together,  and  B  $5  as  often  as  C  $11. 

52.  A  father  and  son  are  aged  respectively  32  and  8. 
How  long  will  it  be  before  the  son  will  be  just  one  half 
the  age  of  the  father  ? 

53.  A  man's  age  was  to  that  of  his  wife  at  the  time 
of  their  marriage  as  4:3,  and  seven  years  after,  their 
ages  were  as  5  :  4.  What  was  the  age  of  each  at  the 
time  of  their  marriage  ? 

54.  One  fifth  of  a  certain  number  minus  one  fourth  of 
a  number  20  less  is  2.     What  is  the  number?     Ans.  60. 

65.  There  are  two  numbers  which  are  to  each  other  as 
J  :  J ;  but  if  9  is  added  to  each,  they  will  be  as  |  :  ^. 
What  are  the  numbers  ?  Ans.  9  and  6. 

56.  A  person  having  spent  $  150  more  than  one  third 
of  his  income  had  $  50  more  than  one  half  of  it  left. 
What  was  his  income  ? 

61.  A  merchant  sold  from  a  piece  of  cloth  a  number 
of  yards,  such  that  the  number  sold  was  to  the  number 
left  as  4  :  5  ;  then  he  cut  off  for  his  own  use  15  yards, 
and  found  that  the  number  of  yards  left  in  the  piece  was 
to  the  number  sold  as  1:2.  How  many  yards  did  the 
piece  originally  contain  ?  Ans.  45. 

58.  Four  places,  A,  B,  C,  and  D,  are  in  a  straight  line, 
and  the  distance  from  A  to  D  is  126  miles.  The  distance 
from  A  to  B  is  to  the  distance  from  B  to  C  as  3  :  4,  and 
one  third  the  distance  from  A  to  B  added  to  three  fourths 
the  distance  from  B  to  C  is  twice  the  distance  from  C  to 
D.  What  is  the  distance  from  A  to  B,  from  B  to  C,  and 
from  C  to  D  ? 

59.  A  laborer  was  hired  for  40  days  ;  for  each  day  he 
wrought  he  was  to  receive  $2.50,  and  for  each  day  he 
was  idle  he  was  to  forfeit  $1.25.  At  the  end  of  the  time 
ho  received  $58.75.     How  many  days  did  he  work? 

Ana    20. 


EQUATIONS   OF   THE   FIRST  DEGREE.',,'  ;'•.;  ;   ''"POJ 

60.  A  cask  which  held  44  gallons  wa^  ,  fi'Ued  xHth-' j^ 
mixture  of  brandy,  wine,  and  water.  There  were  10  gal- 
lons more  than  one  half  as  much  wine  as  brandy,  and  as 
much  water  as  brandy  and  wine.  How  many  gallons 
were  there  of  each  ? 

61.  Two  persons,  A  and  B,  travelling  each  with  $80, 
meet  with  robbers  who  take  from  A  $5  more  than  twice 
what  they  take  from  B;  then  B  finds  he  has  $26  more 
than  twice  what  A  has.     How  much  is  taken  from  each  ? 

Ans.  From  A,  $69  ;  from  B,  $32. 

62.  Four  persons,  A,  B,  C,  and  D,  entered  into  part- 
nership with  a  capital  of  $84816;  of  which  B  put  in 
twice  as  much  as  A,  C  as  much  as  A  and  B,  and  I)  as 
much  as  A,  B,  and  C.     How  much  did  each  put  in  ? 

63.  In  three  cities.  A,  B,  and  C,  1188  soldiers  are  to 
be  raised.  The  number  of  enrolled  men  in  A  is  to  that 
in  B  as  3  :  5  ;  and  the  number  in  B  to  that  in  C  as  8  ;  7. 
How  many  soldiers  ought  each  city  to  furnish  ? 

Ans.  A,  288  ;  B,  480  ;  C,  420. 

64.  Divide  $65  among  five  boys,  so  that  the  fourth 
may  have  $2  more  than  the  fifth  and  $3  less  than  the 
third,  and  the  second  $4  more  than  the  third  and  $5 
less  than  the  first. 

65.  A  merchant  bought  two  pieces  of  cloth,  one  at  tho 
^rate  of   $  5  for  7  yards,  and   the    other  $  2  for  3  yards  ; 

the  second  piece  contained  as  many  times  3  yards  as  the 
first  times  4  yards.  He  sold  each  piece  at  the  rate  of 
$6  for  7  yards,  and  gained  $24  by  the  bargain.  Hov 
many  yards  were  there  in  each  piece  ? 

Ans.  First,  84 ;  second,  63. 

66.  A  drover  had  the  same  number  of  cows  and  sheep. 
Having  sold  17  cows  and  one  third  of  his  sheep,  he  finds 
he  has  three  and  a  half  times  as  many  sheep  as  cows  left. 
How  many  of  each  did  he  have  at  first? 


162  ELKMENTARY   ALGEBRA. 

Ct.  A'  fip.ur  ^dealer  sold  one  fourth  of  all  the  flour  he 
had  and  one  fourth  of  a  barrel ;  afterward  he  sold  one 
third  of  what  he  had  left  and  one  third  of  a  barrel  ;  and 
then  one  half  of  the  remainder  and  one  half  of  a  barrel; 
and  had  15  barrels  left.     How  many  had  he  at  first  ? 

SOLUTION. 

Let  X  =  number  at  first; 

3  X         1 

then  — =  number  after  first  sale, 

4  4 


I— -)  —  s^^^o  —  2^^^  number  after  second  sale, 

d     2(2  —  2)  —  2^^4  —  4^^  number  after  third  sale. 


3 
an ' 


Then  ^  _  !  =,  15 

4  4 

Clearing  of  fractions,       x  —  3  =  60 

Whence  x  =  63,  number  at  first. 

68.  A  merchant  bought  a  barrel  of  oil  for  $50;  at  the 
same  rate  per  gallon  as  he  paid,  he  sold  to  one  man  15 
gallons  ;  then  to  another  at  the  same  rate  two  fifths  of 
the  remainder  for  $  14.  How  many  gallons  did  he  buy 
in  the  barrel  ? 

69.  Two  pieces  of  cloth  of  the  same  length  but  dif- 
ferent prices  per  yard  were  sold,  one  for  S5  and  the^ 
other  for  $1M.  If  there  had  been  5  more  yards  in 
each,  at  the  same  rate  per  yard  as  before,  they  would 
have  come  to  $15.47^f-.  How  many  yards  were  there  iu 
each?  Ans.  21. 

7i(l  A  and  B  began  trade  with  equal  sums  of  money. 
The  first  year  A  lost  one  third  of  his  money,  and  B 
gained  $750.  The  second  year  A  doubled  what  he  had 
at  the  end  of  the  first  year,  and  B  lost  S150,  when  the 
two  had  again  an  equal  sum.  What  did  each  have  at 
first? 


EQUATIONS    OF    THp:   FIRST   DEGREE.,     ,   ;     ,    ,     ,^-08 

Tl.  A  man  distributed  among  his  laborers  ^-2.60  ^p^eQ^, 
and  had  $25  left.  If  he  had  given  each  $'^  as  long  as 
his  money  lasted,  three  would  have  received  nothing. 
How  many  laborers  were  there,  and  how  much  money 
did  he  have?  Ans.  68  laborers,  and  $195. 

T2.  A  man  who  owned  two  horses  bought  a  saddle  for 
$35.  When  the  saddle  was  put  on  one  horse,  their  value 
together  was  double  the  value  of  the  other  horse  ;  but 
when  the  saddle  was  put  on  the  other  horse,  their  value 
together  was  four  fifths  of  the  value  of  the  first  horse. 
What  was  the  value  of  each  horse  ? 

T3.  From  a  cask  two  thirds  full  18  gallons  were  taken, 
when  it  was  found  to  be  five  ninths  full.  How  many 
gallons  will  the  cask  hold  ? 

74.  A  farmer  had  two  flocks  of  sheep,  and  sold  one 
flock  for  $60.  Now  a  sheep  of  the  flock  sold  was  worth 
4  of  those  left,  and  the  whole  value  of  those  left  was  $8 
more  than  the  price  of  8  sheep  of  those  sold,  and  the  flock 
left  contained  40  sheep.  How  many  sheep  did  the  farmer 
sell,  and  what  was  the  value  of  a  sheep  of  each  flock  ? 

Ans.  Number  sold,  15;  value,  $4  and  $1. 

75.  A  man  has  seven  sons  with  2  years  between  the 
ages  of  any  two  successive  ones,  and  the  sum  of  all  their 
ages  is  ten  times  the  age  of  the  youngest.  What  is  the 
age  of  each  ? 

76.  Divide  75  into  two  parts  such  that  the  greater  in- 
creased by  9  shall  be  to  the  less  diminished  by  4  as  3  :  1. 

77.  Divide  a  into  two  parts  such  that  the  greater  in- 
creased by  b  shall  be  to  the  less  diminished  by  <?  a,s  m  :  n. 

78.  What  two  numbers  are  as  3:4,  while  if  8  be 
added  to  each  the  sums  will  be  as  5  :  6  ? 

79.  Divide  127  into  two  parts,  such  that  the  difference 
between  the  greater  and  130  shall  be  equal  to  five  times 
the  difference  between  the  less  and  63. 


ELEMENTAKY  ALGEBRA. 


SECTION   XIY. 

EQUATIONS 

OF    THE    FIEST    DEGREE    CONTAINING    TWO    UNKNOWN 
QUANTITIES. 

109t  Independent  Equations  are  such  as  cannot  be  de- 
rived from  one  another,  or  reduced  to  the  same  form. 

Thus,  a:  +  y  =  10,  I  +  I  =  5,  and  4  X  +  3  iy  =  40  —  y 
are  not  independent  equations,  since  any  one  of  the  three 
can  be  derived  from  any  other  one ;  or  they  can  all  be 
reduced  to  the  form  x  -{-  ^=zlO.  But  x-\-y=:zlO  and 
4:X  =  y  are  independent  equations, 

HO.  To  find  the  value  of  several  unknown  quantities, 
there  must  be  as  many  independent  equations  in  which 
the  unknown  quantities  occur  as  there  are  unknown 
quantities. 

From  the  equation  x -j- y  =  10  we  cannot  determine  the  value 
of  either  ar  or  y  in  known  terms.  If  y  is  transposed,  we  have 
x=  10  —  y;  but  since  y  is  unknown,  we  have  not  determined  the 
value  of  X.  We  may  suppose  y  equal  to  any  number  whatever, 
and  then  x  would  equal  the  remainder  obtained  by  subtracting  y 
from  10.  It  is  only  required  by  the  equation  that  the  sum  of  two 
numbers  shall  equal  10;  but  there  is  an  infinite  number  of  pairs 
of  numbers  whose  sum  is  equal  to  10.  But  if  we  have  also  the 
equation  4  a:  =  y,  we  may  put  this  value  of  y  in  the  first  equation, 
x-{-  y=  10,  and  obtain  a:  -j-  4  a:  =  10,  or  x  =  2 ;  then  4  x  =  8  =  y, 
and  we  have  the  value  of  each  of  the  unknown  quantities. 

ELIMINATION. 
111.    Elimination   is   the   method   of  deriving   from   the 
given  equations  a  new  equation,  or  equations,  containing 
one    (or   more)    less   unknown   quantity.       The   unknown 
quantity  thus  excluded  is  said  to  be  elimiimted. 


EQUATIONS   OF   THE   FIRST   DEGREE.  105 

There  are  three  methods  of  elimination :  — 
I.    By  substitution. 
II.    By  comparison. 

111.  By  combination. 

CASE    I. 

112.  Elimination  by  substitution. 

1.    Given    l^^  +  ^^.^^H,  to  find:r  and  2/. 

OPERATION. 

4x  +  5y=:23  (1)  6x  +  4.y=:   22  (2) 

,  =  ?i^^       (3)       5x  +  4(^A=lf)^  22  (4) 

25x-f  92— 16x  =  110  (5) 

3,=!i=^  =  3   (1)  ■  x=     2  (6) 

Transposing  4  a:  In  (1)  and  dividing  by  5,  we  have  (3),  which 
gives  an  expression  for  the  value  of  y.  Substituting  this  value  of 
y  in  (2),  we  have  (4),  which  contains  but  one  unknown  quantity ; 
i.  e.  y  has  been  eliminated.  Reducing  (4)  we  obtain  (6),  or 
X  =  2.  Substituting  this  value  of  x  in  (3),  we  obtain  (7),  or 
y  =  3.     Hence, 

RULE. 

Find  an  expression  for  the  value  of  one  of  the  unknown 
quantities  in  one  of  the  equations,  and  substitute  this  value 
for  the  same  unknown  quantify  in  the  other  equation. 

Note.  —  After  eliminating,  the  resulting  equation  is  reduced  by 
the  rule  in  Art.  102.  The  value  of  the  unknown  quantity  thus 
found  must  be  substituted  in  one  of  the  equations  containing  the 
two  unknown  quantities,  and  this  reduced  by  the  rule  in  Art.  102. 


Find  the  values  of  x  and  y  in  the  following  equations  :  — 

=  10. 

7. 


2.   Given    |^  +  2/=n|.  Ans.    j^ 

(x  —  y=:    3)  ly 


106 


ELEMENTARY    ALGEBRA. 


3.    Given 


4.   Given 


5.    Given 


9      1^ 


12 


Ans 


+  1-^ 


2x 


x+y 


—  3  =  o) 

—  29  =  0) 


(x=14. 

tyr=15. 

ix=    1. 
l.V  =  22. 


r       ^   I   y_^) 

-j  2  "■     3  ~"  3t5  >■ 

'2j;+  3^  =  2    ) 

(3a:-y=zl6) 


''  — '^^    0 
G.    Given    -<!     2       3 


CASE    II. 
113*   Elimination  by  comparison. 


1. 

Given    |2rJ^  =  2T|'  *°  '^"'^  "^  *°''  y- 

OPEKATION. 

X  - 

-2y  =  6                (1)           2x- 

-3^  =  27 

(2) 

:r  =  6  +  2y     (3) 

•^~         2 

(4) 

6  +  2y=^^  +  ^ 

(5) 

12  +  4y  =  2t  +  y 

(6) 

y=  5 

(T) 

ar=    6+10  = 

16          (8) 

Finding  an  expression  for  the  value  of  x  from  both  (I)  and  (2), 
we  have  (3)  and  (4).  Placing  these  two  values  of  x  equal  to 
each  other  (Art.  13,  Ax.  8),  we  form  (5),  which  contains  but  one 
unknown  quantity.  Reducing  (5)  we  obtain  (7),  or  »/  =  5.  Sub- 
stituting this  value  of  y  in  (3),  we  have  (8),  or  z  =  16.     Hence, 


EQUATIONS   OF   THE   FIRST   DEGREE. 
RULE. 


lOT 


Find  an  expression  for  the  value  of  the  same  unknown 
quantity  from  each  equation,  and  put  these  expressions  equal 
to  each  other. 

By  this  method  of  elimination  find  the  values  of  x  and  y 
in  the  following  equations  :  — 


2.   Given   ■<!  ^  ^  ~  3  "  ^ 
X  -\-y  =  4 


3.    Given 


1  +  5'  =  12  I 


Ans. 


Ans. 


(ic  =  24. 
(v=    0. 


4.   Given    j3-  +  53,  =  2) 

^  =  4l 


6.   Given   ■ 


3 

x-\-y 


3^5 


Ans.    -5  ^       ^' 


Ans.    \ 


=  2. 


7.   Given    i6^-5.y=n> 
(     2x  —  i/  =  ui 


8.   Given   - 


-4-^  = 
4^5 


5 

^-4 


108 


ELEMENTARY   ALGEBRA. 


CASE    III. 
114.   Elimination  by  combination. 

1.   Given    -<  "^  f- ,  to  find  x  and  y. 

X'lx—Zy  —  Z)  ^ 


2x  —  Si/  =  S 


OPEKATION 

Zx  —  2y=    1 

(1) 

6x  — 4y=14 

(3) 

6a:  —  9j^=    9 

(4) 

by—    b 

(5) 

y=  1 

(6) 

2x 


(2) 


(8) 


If  we  multiply  (1)  by  2,  and  (2)  by  3,  we  have  (3)  and  (4), 
in  which  the  coefficients  of  x  are  equal;  subtracting  (4)  from  (3), 
we  have  (5),  which  contains  but  one  unknown  quantity.  Redu- 
cing (5),  we  have  (6),  or  ?/  =  1 ;  substituting  this  value  of  y  in 
(2),  we  obtain  (7),  which  reduced  gives  (8),  or  z  =  3. 


2.    Given 


2-4-     ^ 

2  +  ^=12 
L  3  —  2 


,  to  find  X  aud  y. 


OPERATION. 

!-!=« 

(1) 

-  +  ?-12 
3  —  2 

(2) 

(6) 

=c-|  =  12 

(3) 

9-1=    6 

^-^^ 

(4) 

y=12 

(7) 

x=18 

(5) 

If  we  multiply  (1)  by  2,  we  have  (3),  an  equation  in  which  y  has 
the  same  coefficient  as  in  (2)  ;  since  the  signs  of  y  are  different  in 
(2)  and  (3),  if  we  add  these  two  equations  together,  we  have 
(4),  which  contains  but  one  unknown  quantity.  Reducing  (4),  we 
have  (5),  or  a:  =  18.  Substituting  this  value  of  x  in  (1),  we  have 
(6),  which  reduced  gives  (7),  ory— 12.     Hence, 


EQUATIONS   OF   THE   FIRST    DEGREE. 
RULE. 


109 


Multiply  or  divide  the  equations  so  that  the  coefficients  of 
the  quantity  to  be  eliminated  shall  become  equal ;  then,  if  the 
signs  of  this  quantity  are  alike  in  both,  subtract  one  equa- 
tion from  the  other ;  if  unlike,  add  the  two  equations  to- 
gether. 

Note.  —  The  least  multiplier  for  each  equation  will  be  that  which 
will  make  the  coefficient  of  the  quantity  to  be  eliminated  the  least 
common  multiple  of  the  two  coefficients  of  this  quantity  in  the 
given  equations.  It  is  always  best  to  eliminate  that  quantity  whose 
coefficients  can  most  easily  be  made  equal. 


B}'-  this  method  of  elimination  find  the  values  of  x  and  y 
in  the  following  equations  :  — 


3.  Given    |^-  + 3^-33) 

4.  Given    j   8. +  6,  =  6) 

(lOa:  — 3y  =  4) 


Ans. 


Ans. 


<x=S. 
ix  =  i. 


5.   Given   - 


9    '    6 


12 
5 


Ans 


=  21. 
12. 


6.   Given 


6 

X  —  y 


2 


—  3  =  0 

—  1=0 


1.  Given   - 


2  3  . 

-a:--y=    0 


110  ELEMENTARY   ALGEBRA. 

115.    Find  the  values  of  x  and  y  in  the  following 


Examples. 

Note.  —  Which  of  the  three  methods  of  elimination  should  be 
used  depends  upon  the  relations  of  the  coeflicients  to  each  other. 
That  one  which  will  eUminate  the  quantity  desired  with  the  least 
work  is  the  best. 


1.   Given  j2-+3l/  =  25| 


Ans. 


2.   Given 


5a:  —      y  =r    0 
7a:+?y  =  31 


Ans.  ■<  ^ 

(y=15. 


3.   Given 


y  =  h 


2 

X  —  2  V  ^ 


Ans. 


(v=    1. 


4.   Given 


r— 1  17  ) 

-2-+y=4    f. 
2a:  — 4y=17  ) 


Ans.  ■< 


6.   Given 


1x-\-Zy        1 a:-|-2^-}-3 

3  3  ~  2 

H        2a:— 2y 


3 


=  3 


Ans. 


(a:  =  ll. 
tv=    5. 


6.   Given 


^  I    y 
7  "•     3 


+  -  =  24 
'16         ^   J 


7.    Given 


^-\-y        a:  — y_ 
__ ^^_  _  18 


+  y 


11 


-8H 


EQUATIONS   OF   THE   FIRST   DEGREE. 


Ill 


8.   Given 


5    "•      3    ~~  15 


y  =  T^} 


Ans 


9.   Given 


10.   Given 


bx  + 


54 


=  f  +  4 


Ans. 


-j-  5  y  =  102 


4         '3  2         ^       ^ 


Sx  =  10. 


l"-!;: 


a:=:48. 
28. 


i        1  +  1=    Ol.  Ans.{— _ 

(4;r  —  3v  =  25)  ^ 


=       4. 
3. 


12.   Given 


x  —  y 


b 

x-{-y 


=  2 


Ans, 


(a?  =  a  +  6. 


13. 

Given  < 

14. 

Given  < 

► 

16. 

Given  < 

►  . 

Ans 


=  10. 
20. 


112 


ELEMENTARY   ALGEBRA. 


16. 

Given    s 

^-"^  -X       ^    ' 

• 

n. 

Given    - 

3                        3 

■ 

18. 

Given    < 

[2,       'V=^ 

>• . 

\*--r!,_^      ^ 

19. 

Given    - 

5 

PROBLE 

MS 

PRODUCING  EQUATIONS  OF  THE  FIRST  DEGREE  CON- 
TAINING TWO  UNKNOWN  QUANTITIES. 

116.  Many  of  the  problems  given  in  Section  XIII.  con- 
tain two  or  more  unknown  quantities;  but  in  every  case 
these  are  so  related  to  each  other  that,  if  one  becomes 
known,  the  others  become  known  also  ;  and  therefore 
the  problems  can  be  solved  by  the  use  of  a  single  let- 
ter. But  many  problems,  on  account  of  the  complicated 
conditions,  cannot  be  performed  by  the  use  of  a  single 
letter.  No  problem  can  be  solved  unless  the  conditions 
given  are  sufficient  to  form  as  many  independent  equa- 
tions as  there  are  unknown  quantities. 

1.  A  grocer  sold  to  one  man  T  apples  and  5  pears  for 
41  cents;  to  another  at  the  same  rate  11  apples  and  3 
pears  for  45  cents.     What  was  the  price  of  each  ? 


EQUATIONS  OF   THE  FIRST   DEGREE.  113 

SOLUTION. 

Let  X  =z  the  price  of  an  apple, 


and 

y=    '' 

"      "    a  pear. 

Then,  by  the  conditions. 

7a;  +  5y  =  41 

(1) 

and  11  a;  +    Zy=.    45 

(2) 

55x+15^  =  225 

(3) 

21x4-  15^=123 

(4) 

21+5y  =  41 

a) 

34  a:  =102 

(5) 

y=  4 

(8) 

x=  '  3 

(6) 

We  multiply  (2)  by  5  and  (1)  by  3,  and  obtain  (3)  and  (4)  ; 
subtracting  (4)  from  (3)  we  have  (5),  which  reduced  gives  (6),  or 
a;  =  3.  Substituting  this  value  of  a:  in  (1),  we  have  (7),  which  re- 
duced gives  (8),  or  2/  =  4. 

2.  There  is  a  fraction  such  that  if  2  is  added  to  the 
numerator  the  fraction  will  be  equal  to  ^  ;  but  if  3  is 
added  to  the  denominator  the  fraction  will  be  equal  to  ^. 
What  is  the  fraction  ? 

SOLUTION. 

Let  -  =  the  fraction. 
y 

Then,  by  the  conditions, 

^4-21  /ix  J  ^  1  /n\ 

3x  =  y+3  (3) 

2x  +  4  =  y  (4) 

x  — 4  =  3  (5) 

x=1  (6) 

144-4=18=y       a)  ^  =  ^3  (8) 

Clearing  (1)  and  (2)  of  fractions,  we  obtain  (3)  and  (4)  ;  sub- 
tracting (4)  from  (3),  we  obtain  (5),  which  reduced  gives  (6),  or 
a;  =  7.     Substituting  this  value  of  x  in  (4),  we  have  (7),  or  y=  18. 


114  ELEMENTARY   ALGEBRA. 

3.  There  are  two  numbers  whose  sura  is  28,  and  one 
fourth  of  the  first  is  3  less  than  one  fourth  of  the  second. 
What  are  the  numbers  ?  Ans.  8  and  20, 

4.  The  ages  of  two  persons,  A  and  B,  are  such  that  5 
years  ago  B's  age  was  three  times  A's  ;  but  15  years  hence 
B's  age  will  be  double  A's.     What  is  the  age  of  each  ? 

Ans.   A's,  25  ;  B's,  65. 

6.  There  are  two  numbers  such  that  one  third  of  the 
first  added  to  one  eighth  of  the  second  gives  39 ;  and 
four  times  the  first  minus  five  times  the  second  is  zero. 
What  are  the  numbers  ? 

6.  Find  a  fraction  such  that  if  6  is  added  to  the  nu- 
merator its  value  will  be  ^,  but  if  3  be  added  to  the  de- 
nominator its  value  will  be  ^  ?  *  Ans.  ^j. 

7.  What  are  the  two  numbers  whose  difierence  is  to 
their  sum  as  1:2,  and  whose  sum  is  to  their  product 
as  4  :  3  ? 

SOLUTION. 

Let  X  =  the  greater  and   y  =  the  less. 

Thenx  —  ij:x  +  i/=l:2      (1)  x  +  ?/:xy  =  4:3  (2) 

2x  —  2i/  =  x  +  i/  (3)  Sx  +  3f/  =  4:xy  (4) 

x  =  37/        (5)  9y  +  33^=12/  (6) 

x  =  S  0)  \=.y  (8) 

Having  written  (1)  and  (2)  in  accordance  with  the  statement  in 
the  problem,  we  form  from  them  (3)  and  (4)  by  Art.  106.  Re- 
ducing (3),  we  obtain  (5)  ;  substituting  this  value  of  x  in  (4),  we 
have  (G),  which,  though  an  equation  of  the  second  degree,  can  be 
at  once  reduced  to  an  equation  of  the  first  degree  by  dividing  each 
term  by  y ;  performing  this  division  and  reducing,  we  obtain  (8)  or 
y  =  1 ;  substituting  this  value  of  y  in  (5)  we  obtain  (7),  or  a*  =  3. 


EQUATIONS    OF   THE    FIRST   DEGREE.  '       115 

8.  What  are  the  two  numbers  whose  difference  is  to 
their  sum  as  3  :  20,  and  three  times  the  greater  minus  twice 
the  less  is  35  ? 

9.  There  is  a  number  consisting  of  two  figures,  which 
is  seven  times  the  sum  of  its  figures  ;  and  if  36  is  sub- 
tracted from  it,  the  order  of  the  figures  will  be  inverted. 
What  is  the  number  ?  Ans.  84. 

10.  There  is  a  number  consisting  of  two  figures,  the 
first  of  which  is  the  greater  ;  and  if  it  is  divided  by  the 
sum  of  its  figures,  the  quotient  is  6  ;  and  if  the  order  of 
the  figures  is  inverted,  and  the  resulting  number  divided 
by  the  difference  of  its  figures  plus  4,  the  quotient  will 
be  9.     What  is  the  number  ?  Ans.  54. 

11.  As  John  and  James  were  talking  of  their  money, 
John  said  to  James,  "  Give  me  15  cents,  and  I  shall  have 
four  times  as  much  as  you  will  have  left."  James  said 
to  John,  "  Give  me  7^  cents,  and  I  shall  have  as  much 
as  you  will  have  left."  How  many  cents  did  each 
have  ?  Ans.  John,  45  cents  ;  James,  30  cents. 

12.  The  height  of  two  trees  is  such  that  one  third  of 
the  height  of  the  shorter  added  to  three  times  that  of 
the  taller  is  360  feet ;  and  if  three  times  the  height  of 
the  shorter  is  subtracted  from  four  times  that  of  the  taller, 
and  the  remainder  divided  by  10,  the  quotient  is  17.  Re- 
quired the  height  of  each  tree. 

Ans.  90  and  110  feet. 

13.  A  farmer  who  had  $41  in  his  purse  gave  to  each 
man  among  his  laborers  $2.50,  to  each  boy  $1,  and  had 
$15  left.  If  he  had  given  each  man  $4  and  then  each 
boy  $3  as  long  as  his  money  lasted,  3  boys  would  have 
received  nothing.  How  many  men  and  how  many  boys 
did  he  hire  ? 


116     •  ELEMENTARY   ALGEBRA. 

14.  A  man  worked  10  days  and  his  son  6,  and  they 
received  $31;  at  another  time  he  worked  9  days  and 
his  son  1,  and  they  received  $29.50.  What  were  the 
wages  of  each  ? 

16.  A  said  to  B,  "Lend  me  one  fourth  of  your  money, 
and  I  can  pay  my  debts."  B  replied,  "Lend  me  $100 
less  than  one  half  of  yours,  and  I  can  pay  mine.''  Now 
A  owed  $1200  and  B  $1900.  How  much  money  did 
each  have  in  his  possession  ? 

Ans.  A,  $800  ;  B,  $1600. 

16.  If  a  is  added  to  the  difference  of  two  quantities, 

the  sum  is  b  ;   and  if  the    greater  is  divided  by  the  less, 

the  quotient  will  be  c.     What  are  the  quantities  ? 

.  be  —  ac         ,6  —  a 

Ans. ,      and   -• 

c  —  1  c  —  1 

17.  A  man  owns  two  pieces  of  land.  Three  fourths 
of  the  area  of  the  first  piece  minus  two  fifths  of  the  area 
of  the  second  is  12  acres  ;  and  five  eighths  of  the  area 
of  the  first  is  equal  to  four  ninths  of  the  area  of  the 
second.     How  many  acres  are  there  in  each  ? 

Ans.   Ist,  64  acres  ;  2d,  90  acres. 

18.  A  and  B  begin  business  with  different  sums  of 
money;  A  gains  the  first  year  $350,  and  B  loses  $500, 
and  then  A's  stock  is  to  B's  as  9  :  10.  If  A  had  lost 
$500  and  B  gained  $350,  A's  stock  would  have  been  to 
B's  as  1:3.     With  what  sum  did  each  begin  ? 

Ans.  A,  $1450;  B,  $2500. 

19.  If  a  certain  rectangular  field  were  4  feet  longer 
and  6  feet  broader,  it  would  contain  168  square  feet  more; 
but  if  it  were  6  feet  longer  and  4  feet  broader,  it  would 
contain  160  square  feet  more.  Required  its  length  and 
breadth. 


EQUATIONS   OF   THE   FIRST   DEGREE.  117 

20.  A  market-man  bought  eggs,  some  at  3  for  1  cents 
and  some  at  2  for  5  cents,  and  paid  for  the  whole  $2.62  ; 
he  afterward  sold  them  at  36  cents  a  dozen,  clearing 
$0.62.     How  many  of  each  kind  did  he  buy? 

21.  A  and  B  can  perform  a  piece  of  work  together  in 
12  days.  They  work  together  1  days,  and  then  A  fin- 
ishes the  work  alone  in  15  days.  How  long  would  it 
take  each  to  do  the  work?      Ans.  A  36  and  B  18  days. 

22.  "  I  was  ten  times  as  old  as  you  12  years  ago/' 
said  a  father  to  his  son;  "but  3  years  hence  I  shall  be 
only  two  and  one  half  times  as  old  as  you."  What 
was  the  age  of  each  ? 

23.  If  3  is  added  to  the  numerator  of  a  certain  frac- 
tion, its  value  will  be  §  ;  and  if  4  is  subtracted  from  the 
denominator,  its  value  will  be  J^,     What  is  the  fraction? 

24.  A  farmer  sold  to  one  man  T  bushels  of  oats  and  5 
bushels  of  corn  for  $12.76,  and  to  another,  at  the  same 
rate,  5  bushels  of  oats  and  7  bushels  of  corn  for  $13.40. 
What  was  the  price  of  each  ? 

25.  Find  two  quantities  such  that  one  third  of  the  first 

minus   one  half  the  second   shall   equal   one   sixth   of  a ; 

and  one  fourth  of  the  first  plus  one  fifth  of  the  second 

shall  equal  one  half  of  a.  .         34  a        ,  15  a 

^  Ans.  ^   and  — • 

26.  A  person  had  a  certain  quantity  of  wine  in  two 
casks.  In  order  to  obtain  an  equal  quantity  in  each,  he 
poured  from  the  first  into  the  second  as  much  as  the 
second  already  contained;  then  he  poured  from  the  sec- 
ond into  the  first  as  much  as  the  first  then  contained ; 
and,  lastly,  he  poured  from  the  first  into  the  second  as 
much  as  the  second  still  contained  ;  and  then  he  had  16 
gallons  in  each  cask.  How  many  gallons  did  each  origi- 
nally contain?  Ans.   1st,  22  ;  2d,  10  gallons. 


118 


ELEMENTARY   ALGEBRA. 


SECTION   XV. 

EQUATIONS 

OF  THE  FIRST  DEGREE   CONTAINING  MORE  THAN   TWO 
UNKNOWN  QUANTITIES. 

117.  The  methods  of  elimination  given  for  solving  equa- 
tions containing  two  unknown  quantities  apply  equally 
well  to  those  containing  more  than  two  unknown  quantities. 


(  ^+  y—   ^=  ^) 

1.   Given  • 

J2x  +  3y+4^r=17k 
(Sx  — 2y+  bz=    b) 

OPERATION. 

to  find  X,  y,  and  z. 

x+y— 2=4 

(1)     2a:  +  3.y4-42=l7    (2) 

3ar  — 2?/+    52=    5    (3) 

2x-{-2y  —  2z=    8    (4) 

3ar  +  3?/—    32=  12    (5) 

y  +  62=    9    (G) 

5y-    82=    7    (7) 
5.y  +  302  =  45    (8> 

x+3— 1=4 

(13)               ^  +  6=9     (11) 

382  =  38    (9) 

ar  =  2 

(14)                        y    =  3     (12) 

2=    1(10) 

Multiplying  equation  (1)  by  2  gives  equation  (4),  which  we  sub- 
tract from  (2),  and  obtain  (6)  ;  multiplying  (1)  by  3  gives  (5),  and 
subtracting  (5)  from  (3)  gives  (7).  We  have  now  obtained  two 
equations,  (6)  and  (7),  containing  but  two  unknown  quantities.  Mul- 
tiplying (6)  by  5,  we  obtain  (8),  and  subtracting  (7)  from  (8),  we 
obtain  (9),  which  reduced  gives  2=1.  Substituting  this  value  of 
z  in  (6),  and  reducing,  we  obtain  y  =  3.  Substituting  these  values 
of  y  and  z  in  (1),  and  reducing,  we  obtain  a;  =  2. 


2.   Given 


X  -\-y  =2Q 
y  +z=z2^ 
z  +  w;  =  66 
w)  +  M  =  81 
n  +  X  =  46 


- ,  to  find  u,  w,  X,  y,  and  z. 


EQUATIONS   OF   THE   FIRST    DEGREE.  119 

OPERATION. 

x  +  y  =  26    (1)     y4-z  =  29    (2)     z  +  xv=5Q    (3)     u'  +  j/  =  81    (4)  u  +  x  =  iG    (5) 

y  +  x  =  2Q             z—x=   3             w  +  x=53  u—x  =  2S 


z  —  x=  3    (6)     w  +  x=m    (7)     M  — .T=28    (8)  2x  =  18    (9) 

y  =  n    (11)  Z--12   (12)  IV  ^U  (13)  «  =  37    (14)  a;=   9(10) 

Here  we  subtract  (1)  from  (2),  and  obtain  (G)  ;  then  (6)  from 
(3),  and  obtain  (7);  then  (7)  from  (4),  and  obtain  (8);  then  (8) 
from  (5),  and  obtain  (9),  which  reduced  gives  (10),  or  a:  =  9.  Sub- 
stituting this  value  of  x  in  (1),  (6),  (7),  and  (8),  and  reducing,  we 
obtain  (11),  (12),  (13),  and  (14),  or  ?/  =  17,  2  =  12,  w  =  44,  and 
M=  37. 

Hence,  for  solving  equations  containing  any  number 
of  unknown  quantities, 


RULE. 

From  the  given  equations  deduce  equations  one  less  in 
number,  containing  one  less  unknown  quantity;  and  con- 
tinue thus  to  eliminate  one  unknown  quantity  after  an- 
other, until  one  equation  is  obtained  containing  but  one 
unknown  quantity.  Reduce  this  last  equation  so  as  to  find 
the  value  of  this  unknown  quantity ;  then  substitute  this  value 
in  an  equation  containing  this  and  but  one  other  unknown 
quantity,  and  reducing  the  resulting  equation,  find  the  value 
of  this  second  unknown  quantity ;  substitute  again  these  values 
in  an  equation  containing  no  more  thayi  these  two  and  one 
other  unknown  quantity,  and  reduce  as  before ;  and  so  con- 
tinue, till  the  value  of  each  unknown  quantity  is  found. 

Note.  —  The  process  can  often  be  very  much  abridged  by  the 
exercise  of  judgment  in  selecting  the  quantity  to  be  eliminated,  the 
equations  from  which  the  other  equations  are  to  be  deduced,  the 
method  of  elimination  which  shall  be  used,  and  the  simplest  equa- 
tions in  which  to  substitute  the  values  of  the  quantities  which  have 
been  found. 


120 


ELEMENTARY  ALGEBRA. 


Find   the  values  of  the   unknown  quantities  in  the  fol- 
lowing equations  :  — 


3.   Given 


y  -\-  z  -\-w-\-  u=z  18 
X  -{-  z  -\-w-\-  uz=  17 
X  -\-  y  -\-w-\-  u=  14 
x-\-y-\-z-\-uz=  15 


Note.  —  If  these  equations  are  added  t(^ether  and  the  sum  di- 
vided by  4,  we  shall  have  x-\-y-\-z-\-w-\-  m  =  20;  and  if 
from  this  the  given  equations  are  successively  subtracted,  the  values 
of  the  unknown  quantities  become  known. 


4.   Given 


+  ^y  +  ^2=10 


Ans. 


xz=2. 

y  =  3. 

z  =  Q. 

U=4:. 

w=  5. 


Ans 


rx=z2. 

.]y  =  ^. 
(z=6. 


6.   Given 


2x+  Sy-{-^z  =  Q1 
2y  +  2=:25 


r.=  1. 

An8.-<^y=    1. 

izz=    11. 


6.   Given 


X 

x  +  2 
2x 


2y  —  102=  1  V' 
4y+    3z=  I) 


rx=z  b. 

Ans.  -Jy  =  3. 

(z  =  l. 


7.    Given  -< 


lx  +  ly  +  l.  =  22 

1.1.1 

4^  +  4^+, =^  =  24 

1      .    1      .    I 

x^  +  «y  +  «^=io 


Ans. 


E 


a:  =  20. 

12. 

z  =32. 


EQUATIONS   OF   THE   FIEST   DEGREE. 


121 


8.   Given  < 


-+-  = 


-  +  '  =  - 

y^  X         12 
-  +  '=' 


Note.  —  The  best  method  for  this  example  ia  that  used  in  Ex- 
ample 3,  without  clearing  of  fractions. 


9.   Given 


(    ^+     y+     ^=    6j  fx  = 

42x  +  By  +  4:2  =  20y,  Ans.  -lt/  = 

idx4-1vA-5z=zd2)  (z  = 


x=l. 
2. 
3. 


x  +  iy  =  S1 
10.   Given  -iy  -\-i 


11.   Given 


12.   Given 


13.   Given 


x  +  y  =  a 


rx-f-y  =  ax  rx  = 

<y  +  ^  —  ^r'    Ans.    jy  = 


rabx-\-aby  =  a-\-b  \ 
<acx-{-acz  =  a-\-c  h 
\b  c  y  -\-  b  c  z  =  b   -j-c/ 


1 


x+21  =^y+2S 
9ar  =  2y 


\ 


^(«- 

h+c) 

i(a  +  b-c) 

^  (6  +  c  -  a) 

'    _1 
a 

Ans.    < 

1 

y=-b 

__  1 

.          c 

122  ELEMENTARY  ALGEBRA. 

PROBLEMS 

PRODUCING    EQUATIONS   OF    THE    FIRST    DEGREE    CON- 
TAINING MORE  THAN  TWO   UNKNOWN   QUANTITIES. 

118.  1.  A  merchant  has  three  kinds  of  flour.  He  can 
sell  1  bbl.  of  the  first,  2  of  the  second,  and  3  of  the  third 
for  $85;  2  of  the  first,  1  of  the  second,  and  ^  bbl.  of 
the  third  for  $45.60;  and  1  of  each  kind  for  $41.  What 
is  the  price  per  l>bl .  of  each  ? 

Ans.  1st,  $12;  2d,  $14;  3d,  $15. 

2.  Three  boys.  A,  B,  and  C,  divided  a  sum  of  money 
among  themselves  in  such  a  manner  that  A  and  B  re- 
ceived 18  cents,  B  and  C  14  cents,  and  A  and  C  16.  How 
much  did  each  receive?     Ans.  A,  10  ;  B,  8  ;  C,  6  cents. 

3.  As  three  persons,  A,  B,  and  C,  were  talking  of  their 
ages,  it  was  found  that  the  sum  of  one  half  of  A's  age,  one 
third  of  B's,  and  one  fourth  of  C's  was  33  ;  that  the  sum 
of  A's  and  B's  was  13  more  than  C's  age  ;  while  the  sum 
of  B's  and  C's  was  3  less  than  twice  A's  age.  What 
was  the  age  of  each?     Ans.  A's,  32;  B's,  21  ;  C's,  40. 

4.  As  three  drovers  were  talking  of  their  sheep,  says 
A  to  B,  "If  you  will  give  me  10  of  yours,  and  C  one 
fourth  of  his,  I  shall  have  6  more  than  C  now  has." 
Says  B  to  C,  "If  you  will  give  me  25  of  yours,  and  A 
one  fifth  of  his, 'I  shall  have  8  more  than  both  of  you 
will  have  left."  Says  C  to  A  and  B,  "If  one  of  you 
will  give  me  10,  and  the  other  9,  I  shall  have  just  as 
many  as  both  of  you  will  have  left."  How  many  did 
each  have? 

5.  Divide  32  into  four  such  parts  that  if  the  first  part 
is  increased  by  3,  the  second  diminished  by  3,  the  third 
mnltipliod  by  3,  and  the  fourth  divided  by  3,  the  sum, 
difference,  product,  and  quotient  shall  all  be  equal. 

Ans.  3,  9,  2,  and  18. 


EQUATIONS   OF  THE   FIRST   DEGREE.  123 

6.  If  A  and  B  can  perform  a  piece  of  work  together 
in  8^2-  days,  B  and  C  in  9^^  days,  and  A  and  C  in  8^ 
days,  in   how  many  days  can  each  do  it  alone? 

Ans.  A  in  15,   B  in  18,  and  C  in  21  days. 

7.  Find  three  numbers  such  that  one  half  of  the  first, 
one  third  of  the  second,  and  one  fourth  of  the  third  shall 
together  be  56  ;  one  third  of  the  first,  one  fourth  of  the 
second,  and  one  fifth  of  the  third,  43;  one  fourth  of  the 
first,  one  fifth  of  the  second,  and  one  sixth  of  the  third,  35. 

8.  The  sum  of  the  three  figures  of  a  certain  number  is 
12  ;  the  sum  of  the  last  two  figures  is  double  the  first ; 
and  if  297  is  added  to  the  number,  the  order  of  its  fig- 
ures will  be  inverted.     What  is  the  number? 

Ans.   417. 

9.  A  man  sold  his  horse,  carriage,  and  harness  for 
$150.  For  the  horse  he  received  $25  less  than  five 
times  what  he  received  for  the  harness ;  and  one  third 
of  what  he  received  for  the  horse  was  equal  to  what  he 
received  for  the  harness  plus  one  seventh  of  what  he 
received  for  the  carriage.     What  did  he  receive  for  each? 

Ans.  Horse,  $225;  carriage,  $175;  harness,  $50. 

10.  A  man  -owned  three  horses,  and  a  saddle  which 
was  worth  $45.  If  the  saddle  is  put  on  the  first  horse, 
the  value  of  both  will  be  $  30  less  than  the  value  of  the 
second  ;  if  the  saddle  is  put  on  the  second  horse,  the  value 
of  both  will  be  $55  less  than  the  value  of  the  third  ;  and 
if  the  saddle  is  put  on  the  third  horse,  the  value  of  both 
will  be  equal  to  twice  the  value  of  the  second  minus  $10 
more  than  one  fifth  of  the  value  of  the  first.  What  is 
the  value  of  each  horse  ? 

Ans.    1st,  $100;   2d,  $175;    3d,  $275. 

11.  The  sum  of  the  numerators  of  two  fractions  is  7, 
and  the  sum  of  their  denominators  16  ;  moreover  the  sum 
of  the   numerator  and    denominator  of  the  first  is   equal 


124  ELEMENTARY  ALGEBRA. 

to  the  denominator  of  the  second  ;  and  the  denominator 
of  the  second,  minus  twice  the  numerator  of  the  first,  is 
equal  to  the  numerator  of  the  second.  What  are  the 
fractions  ?  Ans.  f  and  ^. 

12.  A  man  bought  a  horse,  a  wagon,  and  a  harness, 
for  $180.  The  horse  and  harness  cost  three  times  as 
much  as  the  wagon,  and  the  wagon  and  harness  one  half 
as  much  as  the  horse.     What  was  the  cost  of  each  ? 

13.  A  gentleman  gives  $600  to  be  divided  among  three 
classes  in  such  a  way  that  each  one  of  the  best  class  is 
to  receive  $10,  and  the  remainder  to  be  divided  equally 
among  those  of  the  other  two  classes.  If  the  first  class 
proves  to  be  the  best,  each  one  of  the  other  two  classes 
will  receive  $5  ;  if  the  second. class  proves  to  be  the  best, 
each  one  of  the  other  two  classes  will  receive  $4f ;  but 
if  the  third  class  proves  to  be  the  best,  each  one  of  the 
other  two  classes  will  receive  $2.  What  is  the  number 
in  each  class? 

14.  A  cistern  has  3  pipes  opening  into  it.  If  the  first 
should  be  closed,  the  cistern  would  be  filled  in  20  min- 
utes ;  if  the  second,  in  25  minutes  ;  and  if  the  third,  in 
30  minutes.  TTow  long  would  it  take  each  pipe  alone 
to  fill  the  cistern,  and  how  long  would  it  take  the  three 
together  ? 

Ans.  1st,  85f  minutes  ;  2d,  46y\  minutes  ;  3d,  35y\ 
minutes.     The   three  together,    16rfy  minutes. 

15.  Three  men,  A,  B,  and  C,  had  together  $24.  Now 
if  A  gives  to  B  and  C  as  much  as  they  already  have, 
and  then  B  gives  to  A  and  C  as  much  as  the}^  have  after 
the  first  distribution,  and  again  C  gives  to  A  and  B  as 
much  as  they  have  after  the  second  distribution,  they  will 
all  have   the   same   sum.      How   much   did   each   have  at 

.first?  Ans.  A,  $13  ;  B,  $1,  and  C,  $4. 


EQUATIONS   OF   THE   FIKST   DEGREE.  125 

SECTION   XYI. 

POWERS    AND    ROOTS. 

119.  A  Power  of  any  quantity  is  the  product  obtained 
by  taking  that  quantity  any  number  of  times  as  a  factor; 
and  the  exponent  shows  how  many  times  the  quantity  is 
taken  (Art.  24).     Thus, 

a  =z  a}  is  the  first  power  of  a  ; 
a  a^=zcP'       "       second  power,  or  square,  of  a  ; 
«  «  rt  =  a^       "       third  power,  or  cube,  of  a  ; 
a  a  a  a:=.a^       "       fourth  power  of  a  ; 
and  so  on, 

120.  In  order  to  explain  the  use  of  negative  indices, 
we  form,  by  the  rules  of  division,  the  following  series:  — 


a\ 

a\ 

a\ 

a\ 

a, 

1, 

1 
a' 

1 

1 
a'' 

1 

a*' 

1 

«^ 

a\ 

a\ 

a\ 

a\ 

a\ 

a-\ 

a-\ 

«-^ 

a-^ 

a-\ 

We   form   the  first  series  as   follows:    a^  divided   by   a  gives  a*,* 
a*  by  a,  gives  a^ ;   a?  by  a,  gives  a" ;   a^  by  a,  gives  a ;  a  by  a,  gives 

1 ;   1  by  a,  gives  - ;    -  by  a,  gives  -  j ;    — ^  by  a,  gives  — ^ ,  and  so  on. 

The  second  series  is  formed  in  the  same  way  from  a^  to  a;  but  if 
we  follow  the  same  rule  of  division  from  a  toward  the  right  as  from 
a^  to  a,  viz.  subtracting  the  index  of  the  divisor  from  that  of  the  div- 
idend^ a  divided  by  a,  gives  a" ;  cP  by  a,  gives  a-^ ;  read  a,  with  the 
negative  index  one ;  a~^  by  a,  gives  a~^ ;  a~^  by  a,  gives  a~^  \  and 
so  on. 

From  this  we  learn, 

Ist.   TJmt  the  0  power  of  every  quantity  is  1  ; 
2d.   That  a~\  a~^,  a~^,  &c.,   are  only  different  ways  of 
writmq  -,     „,    -,,  dec. 


126  ELEMENTARY  ALGEBRA. 

Any  two  quantities  at  equal  distances  on  opposite  sides 
of  a°,  or  1,  are  reciprocals  of  each  other. 

121 1  The  rules  given  for  the  multiplication  and  divis- 
ion of  powers  of  the  same  quantity  (Arts.  50  and  54) 
apply  equally  well  whether  the  exponents  are  positive 
or  negative.     For 

a«    -?-  a-2  =  a^  -h   \  =  a«  X  «'  =  a' 

a"'  -T-  a"^    z=  -.  -i-  a*  =  -n*    or  a~" 
a'  a" 

The  following  examples  m  multiplication  are  to  be  done 
according  to  the  rules  for  the  multiplication  of  powers  of 
the  same  quantity  by  each  other,  given  in  Art.  50  ;  and 
those  in  division,  by  the  rule  for  the  division  of  powers 
of  the  same  quantity  by  each  other,  given  in  Art.  54. 

1.  Multiply  x'  by  x~^.  Ans.  x^. 

2.  Multiply  a*  by  a~^. 

3.  Multiply  a?*  by  x'^.  Ans.  ar^  or  1. 

4.  Multiply  y''  by  y*. 

5.  Multiply  a-*x^f  by  a'^x-^f/'^. 

Ans.  a-^aPip,  or  ar^y"'. 

6.  Multiply  4:X~^y-^z  by  ^x^^z^. 

7.  Multiply  llx'^y^z-'^  by  4  ar-^y"-*  z"*. 

8.  Multiply  ""iLtSll  by  5  a*  b-^  c\ 

o 

9.  Divide  x^  by  a;"".  Ans.  x**. 

10.  Divide  x^  by  x~''. 

11.  Divide  a:"®  by  x~\  Ans.  x'*. 


EQUATIONS    OF   THE   FIRST   DEGREE.  127 

12.  Divide  y-^  by  y^. 

13.  Divide  y''^  by  y'"^.  Ans.  y'^. 

14.  Divide  a-^hc^  by  d^}r^c~'^.  Ans.  ar^Wc^. 

15.  Divide  l^x'y-^z  by  4a:2y-'^23 

16.  Divide  4  o^"^ y-^  ;^  by  2  «  x-'^ y-^  z^. 

17.  Divide  laHx-^y^  by   1 0  a  J-^  x*  ^"^  ;22 

18.  Divide  Ui:  a"^  b  c-'^  x"- y'^  z  by  16  a^J-^c-^a^^/. 

122.  It  follows  from  the  preceding  article  that  a  factor 
may  he  transferred  from  the  numerator  of  a  fraction  to  its 
denominator,  or  vice  versa,  provided  the  sign  of  the  expo- 
nent of  the  factor  is  changed  from  -\-  to  — ,  or  —  to  -{-.     For 


I    r=a«X^.  =  «'X^-^ 


X 


gf'      1  5_1      ,       1    1_^       _5  1 

-  — -     ^^~y~^~y    '    ^      ~ar^y 

—  =  '    X  i  =  — 

X  dJ        X         dJ  X 

1 .   Transfer  the  denominator  of  -^—^ — r  to  the  numerator. 

o  c^  y-^ 


2.   Transfer  the  numerator  of  -i-„ — ,  to  the  denominator. 


Ans.  a!  h~^  c~'^  x^  y . 

,6 
^4  ^2  2-1 


Ans. 


2  .V— I 


^2  J-7  ^-6  ^  2/' 

3.  Transfer  the  denominator  of  — ~^  to  the  numerator. 

a  x^  M — *  2 

4.  Transfer  the  numerator  of  — r^-j—  to  the  denominator. 

oca 


128  ELEMENTABY  ALGEBRA. 

6.   Free  from  negative  exponents       _^  .    _4- 

Ans.    , — :, 

7  acrxy 

6.    Free  from  negative  exponents  — 3— j— j- 

T.    Free  from  negative  exponents  — ^    i^^^  — . 

Ans. 


8.    Free  from  negative  exponents  j '  :^_,  .     ' 


(x  —  y)-*  (x -{- y) 

us.  ^;~^> 

(a^  +  y) 


9.    Free  from  negative  exponents  -^rK^~m ' 

INVOLUTION. 

123.  Involution  is  the  process  of  raising  a  quantity  to 
a  power. 

124.  A  quantity  is  involved  by  taking  it  as  a  factor  as 
many  times  as  there  are  units  in  the  index  of  the  re- 
quired power. 

125.  According  to  Art.  48, 

(+«)X(+o)  =  +  <»'. 
(+«)  X  (+a)  X  (+a)  =  (+a^)  X  (+«)  =  +  a'. 
and  so  on ; 

and  (—0  )  X  (—a)  =  -|-  a^, 

(—a)  X  (—a)  X  i-a)  =  (+«')  X  {-a)  =  —  a\ 
(—  a)  X  (— a)  X  (-a)  X  (— «)  =  (—a')  X  (-a)  =  +  aS 
and  so  on. 

Hence,  for  the  signs  we  have  the  following 

RULE. 
0/  a  positive  quantity  all  the  powers  are  positive. 
Of  a  negative  quantity  the  even  powers  are  positive,  and 
the  odd  powers  negative. 


INVOLUTION.  129 

INVOLUTION    OF    MONOMIALS. 

126.   To  raise  a  monomial  to  any  required  power. 

1.  Find  the  third  power  of  2  a^  b. 

OPERATION. 

(2  aHy  =  2anx2anx2aH  (1) 

=  2.  2.  2.  a^  a^  a^hhh  (2) 

r=:  8  aH^  (3) 

According  to  Art.  1 24,  to  raise  2  a^  6  to  the  third  power  we  take 
it  as  a  factor  three  times  (1)  ;  and  as  it  makes  no  diflference  in  the 
product  in  what  order  the  factors  are  taken,  we  arrange  them  as 
in  (2);  performing  the  multiplication  (Art.  50)  expressed  in  (2), 
we  have  (3).     Hence, 

RULE. 

Multiply  the  exponent  of  each  letter  by  the  index  of  the 
required  power,  and  prefix  the  required  power  of  the  nu- 
merical coefficient,  remembering  that  the  odd  powers  of  a 
negative  quantity  are  negative,  while  all  other  powers  are 
positive. 

Note.  —  It  follows  that  the  power  of  the  product  is  equal  to  the 
product  of  the  powers. 

2.  Find  the  square  of  2  x.  Ans.  4  x^. 
3.-  Find  the  cube  of  3a?2.                                 Ans.  21  x\ 

4.  Find  the  fourth  power  of  a^  W.  Ans.  a^^  h^, 

5.  Find  the  third  power  of  4a^a;.  Ans.  64  a®  a:^. 

6.  Find  the  square  of  2  x~^.  Ans.  4  x~'^. 

7.  Find  the  cube  of  '^x'^y^.  Ans.  27  x"^/. 

8.  Find  the  with  power  oi  ah.  Ans.  a'"5"'. 

9.  Find  the  third  power  of  —  3  a^  h.  Ans.  —  27  a^  i^ 


130  ELEMENTARY  ALGEBRA. 

10.  Expand  {—2a^xy.  Ans.  16  a^^a:\ 

11.  Expand  (Sa^ir)"',  Ans.  S'^a^'"^^". 

12.  Expand  (2x^i/)\ 

13.  Expand  (—4:a^x'')\ 

14.  Expand  {-^Sx^y)\ 

15.  Expand  (—  a'^.  Ans.  —  ar^. 

16.  Expand  (x-^y^y. 

17.  Expand  ( — 4x"^^)^  Ans.  — |-. 

18.  Expand  (Sa^'x^y. 

19.  Expand  (— 2a:-8^-")». 

20.  Expand  (— 3  a^-^"  y'"/. 

21.  Expand  (— 9a-^6-''^a;2^'*)». 

INVOLUTION    OF    FRACTIONS. 
127.   To  involve  a  fraction. 
1.   Find  the  cube  of 


V2  6/   ~  2  i 


26 
OPERATION.  According  to  Art.  124, 

6  '^  2^  ^  26        8^  quantity  we  must  take  it 

three  times  as  a  factor; 
taking  —7  three  times  as  a  factor,  and  performing  the  multiplica- 
tion by  Art.  95,  we  have  — r:.     Hence, 


RULE. 

Involve  both  numerator  and  denominator  to  the  required 
power. 


INVOLUTION.  131 

2  a  4  a* 

2.  Find  the  square  of  —5.  Ans.  ^• 

3.  Fmd  the  cube  of — zr-o-,-  Ans.  — 


21  &d? 


4.    Find  the  fourth  power  of  "  _^ 


6.    Find  the  fifth  power  of  —  ^^'. 

—  2  a  a;-^ 


6.    Find  the  third  power  of 
t.   Find  the  mth   power  of 


3  x-'"  y^ 


2  c?  x~^ 

8.  Find  the  fourth  power  of  — ,—^ — ^,-. 

3  a^  6-*  C-* 

9.  Find  the  third  power  of  —  ^  m^ dr^\' 

0  X      'IT  Z 

10.  Find  the  fourth  power  of  —  _      o"  ,    " 

2  a;~^  w^  z 

11.  Find  the  fifth  power  of  —  o^-Tpas* 


INVOLUTION    OF    BINOMIALS. 

128.  A  BINOMIAL  can  be  raised  to  any  power  by  suc- 
cessive multiplications.  But  when  a  high  power  is  re- 
quired, the  operation  is  long  and  tedious.  The  Binomial 
Theorem,  first  developed  by  Sir  Isaac  Newton,  enables  us 
to  expand  a  binomial  to  any  power  by  a  short  and  speedy 
process. 

129.  In  order  to  investigate  the  law  which  governs  the 
expansion  of  a  binomial  we  will  expand  a  -\-  b  and  a  —  ^ 
to  the  fifth  power  by  multiplication. 


132  ELEMENTAEY  ALGEBBA. 


a  +6 

a  +b 

a^  + 

a6 

a6  +, 

&» 

a2  4-2 

a6  + 

6^ 

. 

• 

•                  • 

a  +h 

, 

a«  +  2 

an  + 

«6« 

a'b-^- 

2aJ''  + 

6^ 

a«  +  3 

an-\- 

3a62  +, 

. 

«  +6 

a* +  3 

a«6  + 

3  a^  ^2  _|. 

aW 

an-\- 

3  a2  //-  + 

3 

aV" 

+  i' 

a^  +  4 

aH  + 

Qan^-\- 

4 

ah^ 

+  i* 

. 

«  +i 

a^  +  4^ 

a^5  + 

6a«^  + 

4 

aH^ 

+  « 

i* 

a*6  + 

4a«62  4. 

6 

dn^ 

+  4a 

i« 

+  ¥ 

2d  power. 


3d  power. 


4th  power. 


a5  -^  5  an  +  10  aH2  +  10  a2  63  4-  5  a  ^.*  +  b^       5th  power. 


a  —  h 
a  —  6 
o^  —     aft 

a^  —  2ah  -\-h'^ 2d  power 

a  —  h 

df:^2  a'h-\-       ^ 


a    u  -|—  a  u 

—  g'^  6  +    2  g  6-^  —  6» 

a8  — 3g26+    3  a 6'^  —  6«         ....     3d  power. 
a  —  h 

a*  _  4  ^8  ^  _|_    6  g2  62  —    4  a  63^>  .    4th  power. 

a  —b 

ati^^a*b-{'    QaH'^—    4  a'^"  +     ab* 

—  a*b+    4taH^—    6aH''-^^ab*  —  5» 

o8  _  6  a*  6  +  10  a8  6'^  —  10  o^  b^~^5^b*~^^^^^      6th  power 


INVOLUTION.  183 

By  examining  the  different  powers  of  a  -\-  h  and  a  —  6 
in  these  Examples,  we  shall  find  the  following  invariable 
laws  governing  the  expansion  :  — 

1st.  The  leading  quantity  (i.  e.  the  first  quantity  of  the 
binomial)  begins  in  the  first  term  of  the  power  with  an  ex- 
ponent equal  to  the  index  of  the  power,  and  its  exponent 
decreases  regularly  by  one  in  each  successive  term  till  it  dis- 
appears ;  the  following  quantity  {i.  e.  the  second  quantity 
of  the  binomial)  begins  in  the  second  term  of  the  power  with 
the  exponent  one,  and  its  exponent  increases  regularly  by 
one  till  in  the  last  term  it  becomes  the  same  as  the  index  of 
the  power. 

Thus,  in  the  fifth  power  the 

Exponents  of  a  are     5,     4,     3,     2,     1. 
Exponents  of  b  are  1,     2,     3,     4,     6. 

It  will  be  noticed  that  the  sum  of  the  exponents  of  the 
letters  in  any  term  is  equal  to  the  index  of  the  power. 

2d.  The  coefficient  of  the  first  term  is  one ;  of  the  second, 
the  same  as  the  index  of  the  power ;  and  universally,  the  co- 
efficient of  any  term,  multiplied  by  the  exponent  of  the  lead- 
ing quantity,  and  this  product,  divided  by  the  exponent  of 
the  following  quantity  increased  by  one,  will  give  the  co- 
efficient of  the  succeeding  term. 

Thus,  in  the  fifth  power,  5,  the  coeflScient  of  the  second 
term,  multiplied  by  4,  a's  exponent,  and  divided  by 
1  plus  1,  6's  exponent  plus    1,   =  —— —  =  10,  the    coeflfi- 

cient  of  the  third  term. 

The  coefficients  are  repeated  in  the  inverse  order  after 
passing  the  middle  term  or  terms,  so  that  more  than  half 
of  the  coefficients  can  be  written  without  calculation.  The 
number  of  terms  is  always  one  more  than  the  index  of 


134  ELEMENTARY   ALGEBRA. 

the  power ;  i.  e.  the  second  power  has  three  terms  ;  the 
third  power,  four  terras  ;  and  so  on.  When  the  number 
of  terms  is  even,  i.  e.  when  the  index  of  the  power  is 
odd,  the  two  central  terms  have  the  same  coefiBcient. 

3d.  When  both  terms  of  the  binomial  are  positive,  all  the 
terms  of  the  power  are  positive;  but  when  the  second  term 
is  negative,  tlwse  terms  which  contain  odd  poivers  of  the 
following  quantity  are  negative,  and  all  the  others  positive; 
or  eveiy  alternate  term,  beginning  with  the  second,  is  negor 
tive,  and  the  others  positive. 

1.  Expand  {x  -\-  yY. 

OPERATION. 

According  to  the  law,  the  first   term  will  be  a:", 

and  the  second  term  -|"  ^  ^  y* 

4 

The  coefficient  of  the  third  term  will  be  — ^—  , 

and  the  third  term  -\-2^j?i^. 

2 

28  \x  ■«. 

The  coefficient  of  the  fourth  term  will  be  — - — , 

and  the  fourth  term  -\-  b^3?t^, 

14 

The  coefficient  of  the  fifth  term  will  be  — P — i 

and  the  fifth  term  70a^y*. 

Having  found  the  preceding  coefficients  and  the  coefficient  of  the 
middle  term,  we  can  write  the  others  at  once.     Hence, 

(X  +  y)8  =  xs  +  8a:7y  +  28a:«y2  +  56a:5y3  4.  TOx^y*  +  56x^y5  .j.  28i2y«  +  %xy^  -f  y8. 

2.  Expand  {a  —  h)\ 

Ans.  a«— 6a^H-15«**"— 20o«6«+16a2i*— 6a^+6«. 

3.  Expand  (m  +  ny. 

4.  Expand  {b  -r-  yY. 
6.    Expand  {a  —  xf^. 


INVOLUTION.  135 

6.    Expand  {b -+- cY^. 
1.    Expand  (x  +  1)^ 

Note.  —  Since  all  the  powers  of  1  are  1,  1  is  not  written  when 
it  appears  as  a  factor;  but  its  exponent  must  be  used  in  obtaining 
the  coefficients. 

Ans.  x^  -{-  b  x^  +  10  x''  -\-  10  x''  +  5  X  +  I. 

8.  Expand   (1  —  y)^ 

Ans.   1  —  6y  +  15  f  ~  20/  +  15y*  —  6/  +  /. 

9.  Expand  (a  —  1)^ 

130.  When  the  terms  of  the  binomial  have  coejfficients 
or  exponents  other  than  1,  the  theorem  can  be  made  to 
apply  by  treating  each  term  as  a  single  literal  quantity. 
In  the  expansion,  each  factor  should  be  enclosed  in  a 
parenthesis,  and  after  the  expansion  of  the  binomial  by 
the  binomial  theorem,  the  work  should  be  completed  by 
the  expansion  of  the  enclosed  factors,  according  to  the 
rule  for  the  expansion  of  monomials. 

1.  Expand  (2  x  —  y^)*. 

OPERATIOX. 

(2  xy  -  4  (2  xy  (f)  +  6  (2  xy  (fy  _  4  (2  a.)  (/)« +  (fy 

Expanding  each  factor  as  indicated,  we  have 

* 

16  x^  —  32  x^f  +  24  x'^y^  _  8  x/  +  / 

2.  Expand  (Sx^—  2yy. 

(3a:2)5  _  5 (3 a:2)4  (2y)  +  10  (3x2)3  (2^)2  _  IQ  (3x2)2  (2y)3  -|-  5 (3ar2)  (2y)4  —  (2^)5. 
Ans.  243x^''  —  8103^y-{-1080x^f—720xUf-\-24:03^y*  —  32y^. 

Note.  —  Any  letters,  as  a  and  6,  might  be  substituted  for  3  a;' 
and  2  y,  and  the  expansion  of  (a  —  by  Written  out,  and  then  the 
values  of  a  and  b  substituted. 

3.  Expand  (a^  _  3  by. 

Ans.  a«— 12a«i  +  54a^62_  lOS  a^  b^ -\- SI  bK 

4.  Expand  (x^  —  fy. 


136  ELEMENTARY   ALGEBEA. 

5.  Expand  (2  a  +  T)«. 

Ans.    8  a«  + 84^2 +  294  a +  343. 

6.  Expand  {2  a  c  —  a:)*. 

Ans.   16aV*  — 32aVa:  +  24a2c2;r2  — 8acx»  +  a:*. 

7.  Expand  (a^a;  — 2^f. 


^-•1^  +  1  +  ^  +  2-^  +  -*. 


8.  Expand  (s  +  ^)  • 

9.  Expand  (I- ly. 

32  48~  "1        36~  54      ^    162  243 

10.   Expand  (|  —  lY. 

U.   Expand  (V^-L^)'. 

,         8i»          .       .     9:i:j/«        27.v» 
Ans.  — -r'y-l-    -^ ^. 

12.  Expand  (i  +  iy. 

Ans.  ^  +  ^^  +  ^  ^i?+F 

13.  Expand  (ac — -J.  « 

14.  Expand  (x-{--\. 

15.  Expand  (l  —  -V. 

16.  Expand  ^2  a^  _  IV. 

17.  Expand  0  +  iV. 

18.  Expand  (i--^. 


EVOLUTION.  137 

131.  The  Binomial  Theorem  can  be  applied  to  the  ex- 
pansion of  a  polynomial.  Thus,  in  a  -]-  b  —  c,  a  -{-  b 
can  be  treated  as  a  single  term,  and  the  quantity  can  be 
written  (a  -\-  h)  —  c.  In  like  manner,  a  -\-  b  -\-  x  —  y  can 
be  written  (a -\- b)  -\-  (x  —  y).  In  such  cases  it  is  easier 
to  substitute  a  single  letter  for  the  enclosed  terms,  and 
after  the  expansion  to  substitute  the  proper  values. 

1 .  Expand  (a -\-  b  —  c)^. 

OPERATION. 

Put  a  -\-  b  =  X 

{x  —  cy  =  x^  —  Sx^c-[-Sxc^  —  c^ 
Substituting  for  x,  its  value,  a  -\-  b', 

(a  +  6  — cP  =  a3  +  3o26  +  3a62  +  63_3a2c  — 6a6c  — 362  c  +3ac3  +  36c2  — c3 

2.  Expand  (2  a  —  b  —  c  —  dy. 

Note.  —  For  2  a  —  b  —  c  —  d  write  (2  a  —  b)  —  (c  -}-  d). 

Ans.  Aa^—Aab-\-W—Aac—Aad-\-2bc  +2 W-[-c2+2cc/-f  d^. 

3.  Expand  {^x  —  ^y  —  a-^  bf. 

4.  Expand  {lx  —  a-{-bf, 

EVOLUTION. 

132.  Evolution  is  the  process  of  extracting  a  root  of 
a  quantity.     It  is  the  reverse  of  Involution. 

133.  A  ROOT  of  any  quantity  is  a  quantity''  which  taken 
as  a  factor  a  given  number  of  times  will  produce  the 
given  quantity. 

The  number  of  times  the  root  is  to  be  taken  as  a  factor 
depends  upon  the  name  of  the  root.  Thus,  the  second 
or  square  root  of  a  quantity  is  a  quantity  which  taken 
twice  as  a  factor  will  produce  the  given  quantity  ;  the 
third  or  cube  root  is  a  quantity  which  taken  three  times 
as  a  factor  will  produce  the  given  quantity  ;  and  so  on. 


138  ELEMENTARY    ALGEBRA. 

A  Root  is  indicated  by  the  radical  sign  \/,  or  by  a 
fractional  exponent.     Thus, 

t^  X,   or  a:^  indicates  the  square  root  of  x. 
y/lc,  or  a:^  "  "     cube         "     "     " 

V^x,    or  J^  "  "     mth  "     "    " 

134,  A  root  and  a  power  may  be  indicated  at  the  same 
time.  Thus,  v^x*,  or  x^,  indicates  the  cube  root  of  the 
fourth  power  of  x,  or  the  fourth  power  of  the  cube  root 
of  X  ;  for  a  power  of  a  root  of  a  quantity  is  equal  to  tlw 

i 

same  root  of  the  same  power  of  the  quantity.  -^8=^  or  8^ 
is  the  square  of  the  cube  root  of  8,  or  the  cube  root  of 
the  square  of  8,  i.  e.  4. 

135»  A  perfect  power  is  a  quantity  whose  root  can  be 
found.  A  perfect  square  is  one  whose  square  root  can 
be  found  ;  a  perfect  cube  is  one  whose  cube  root  can  be 
found ;  and  so  on. 

136*  Since  Evolution  is  the  reverse  of  Involution,  the 
rules  for  Evolution  are  derived  at  once  from  those  of 
Involution.  And  therefore,  as  according  to  Art.  125  an 
odd  power  of  any  quantity  has  the  same  sign  as  the 
quantity  itself,  and  an  even  power  is  always  positive, 
we  have  for  the  signs  in  evolution  the  following 

RULE. 

An  odd  root  of  a  quantity  has  the  same  sign  as  the  quan- 
tity itself 

An  even  root  of  a  positive  quantity  is  either  positive  or 
negative. 

An  even  root  of  a  negative  quantity  is  impossible,  or  im- 
aginary. 


EVOLUTION.  139 

SQUARE    ROOT    OF    NUMBERS. 

137i  The  Square  Root  of  a  number  is  a  number  which, 
taken  twice  as  a  factor,  will  produce  the  given  number. 

138.  The  square  of  a  number  has  twice  as  many  figures 
as  the  root,  or  one  less  than  twice  as  many.     Thus, 

Roots,         1,  10,  100,  1000. 

Squares,      1,  100,  10000,  1000000. 

The  square  of  any  number  less  than  10  must  be  less  than  100; 
but  any  number  less  than  10  is  expressed  by  one  figure,  and  any 
number  less  than  100  by  less  than  three  figures;  i.  e.  the  square  of 
a  number  consisting  of  one  figure  is  a  number  of  either  one  or  two 
figures.  The  square  of  any  number  between  10  and  100  must  be 
between  100  and  10000;  i.  e.  must  contain  more  than  two  figures 
and  less  than  five.  And  the  square  of  any  number  between  100 
and  1000  must  contain  more  than  four  figures  and  less  than  seven. 

Hence,  to  ascertain  the  number  of  figures  in  the  square 
root  of  a  given  number, 

Beginning  at  units,  point  off  the  number  into  periods  of 
two  figures  each ;  there  will  be  as  many  figures  in  the  root 
as  there  are  periods,  and  for  the  incomplete  period  at  the 
left,  if  any,  one  more. 

139.  To  extract  the  square  root  of  a  number. 
1.    Find  the  square  root  of  5329. 

From  the  preceding  explanation,  it  is  evident  that  the  square 
root  of  5329  is  a  number  of  two  figures,  and  that  the  tens  figure 
of  the  root  is  the  square  root  of  the  greatest  perfect  square  in  53 ; 
i.  e.  v/49,  or  7.  Now,  if  we  represent  the  tens  of  the  root  by  a 
and  the  units  by  b,  a  -\-  b  will  represent  the  root ;  and  the  given 
number  will  be 

(a -\- by  ==  a^ -{- 2  a  b -\- b\ 

Now  a"  ==  70^  =  4900  ; 

therefore,  2  a  6  -[-  6^  =  5329  —  4900  =  429. 

But  2ab-\-  W  ={2a-\-b)b; 


140  ELEMENTARY   ALGEBRA. 

If  therefore  429  is  divided  by  2  a  -|-  h,  it  will  give  h  the  units  of 
the  root.  But  b  is  unknown,  and  is  small  compared  with  2  a ; 
we  can  therefore  use  2  a  =  140  as  a  trial  divisor.  429  -^  140, 
or  42-J-14  =  3,  a  number  that  cannot  be  too  small  but  may  be 
too  great,  because  we  have  divided  by  2  a  instead  of  2  a  -f-  &. 
Then  6=3,  and  2  a  -f  6  =  140  -j-  3  =  143,  the  true  divisor ;  and 
(2  a  -f  6)  5  =  143  X  3  =  429  ;  and  therefore  3  is  the  unit  figure  of 
the  root,  and  73  is  the  required  root.  The  work  will  appear  as 
follows ;  — 


OPERATION. 

5  3  2  9  (7  3 

a^TO 

49 

b=     3 

2a  +  6  =  14  3)429 
(2a-\-h)h=  429 

Hence,  to  extract  the  square  root  of  a  number, 

RULE. 

Separate  the  given  number  into  periods  of  two  figures  each, 
by  placing  a  dot  over  units,  hundreds,  &c. 

Find  the  greatest  square  in  the  left-hand  period,  and  place 
its  root  at  the  right. 

Subtract  the  square  of  this  root  figure  from  the  left-hand 
period,  and  to  the  remainder  annex  the  next  peHod  for  a 
dividend. 

Double  the  root  already  found  for  a  trial  divisor,  and, 
omitting  the  right-hand  figure  of  the  dividend,  divide,  and 
place  the  quotient  as  the  next  figure  of  the  root,  and  also  at 
the  rigid  of  the  trial  divisor  for  the  true  divisor. 

Multiply  the  true  divisor  by  this  new  root  figure,  subtract 
the  product  from  the  dividend,  and  to  the  remainder  annex 
the  next  period,  for  a  new  dividend. 

Double  the  part  of  tlie  root  already  found  for  a  trial  di- 
visor, and  proceed  as  before,  until  all  tlie  penods  have  been 
employed. 


EVOLUTION.  141 

Note  1.  —  When  a  root  figure  is  0,  annex  0  also  to^the  trial  di- 
visor, and  bring  down  the  next  period  to  complete  the  new  dividend. 

Note  2.  —  If  there  is  a  remainder,  after  using  all  the  periods  in 
the  given  example,  the  operation  may  be  continued  at  pleasure  by 
annexing  successive  periods  of  ciphers  as  decimals. 

Note  3.  —  In  extracting  the  root  of  any  number,  integral  or  deci- 
mal, place  the  first  point  over  unit's  place ;  and  in  extracting  the 
square  root,  over  every  second  figure  from  this.  If  the  last  period  in 
the  decimal  periods  is  not  full,  annex  0. 

2.    Find  the  square  root  of  46225. 


OPERATION. 


We  suppose  at  first  that  a  rep- 
resents the  hundreds  of  the  root, 
46225  (2  15  and  b  the  tens ;  proceeding  as  in 

4  Ex.  1,  we  have  21  in  the  root. 

A     \  A  o  Then    letting    a    represent    the 

'  hundreds  and  tens  together,  i.  e. 

A    1  o  ' 

21  tens,  and  b  the  units,  we  have 

4  2  5)  2  1  2  5  2  a,   the   2d   trial   divisor,  =  42 

2  12  5  tens ;  and  therefore  6^5;  and 

2  a  -f  6  =  425 ;  and  215  is  the 
required  root. 

3.    Find  the  square  root  of  5013.4. 

operation. 

50  1  3.4  6(7  0.80  5-1- 

49 

140.8)113.40 
112.64 


141.605)  .760000 
.Y08025 

4.  Find  the  square  root  of  288369.  Ans.  537. 

5.  Find  the  square  root  of  42849.  Ans.  207. 

6.  Find  the  square  root  of  173.261.  Ans.   13.16-f. 

7.  Find  the  square  root  of  .9.  Ans.   .948-t-. 


142  ELEMENTARY   ALGEBRA. 

8.  Fini^  the  square  root  of  2.  Ans.  1.4l42-f-. 

9.  Find  the  square  root  of  484. 

10.  Find  the  square  root  of  48.4. 

11.  Find  the  square  root  of  .064. 

12.  Find  the  square  root  of  .00016. 

Note.  —  As  a  fraction  is  involved  by  involving  both  numerator  and 
denominator  (Art.  127),  the  square  root  of  a  fraction  i.s  the  square  root 
of  the  numerator  divided  by  the  square  root  of  the  denominator. 

13.  What  is  the  square  root  of  |  ?  Ans.  §. 

14.  What  is  the  square  root  of  ^|  ? 

15.  What  is  the  square  root  of  ^^j  ?     -^^:=^^.     Ans.  f. 

Note.  —  If  both  terms  of  the  fraction  are  not  perfect  squares, 
and  cannot  be  made  so,  reduce  the  fraction  to  a  decimal,  and  then 
find  the  square  root  of  the  decimal.  A  mixed  number  must  be  re- 
duced to  an  improper  fraction,  or  the  fractional  part  to  a  decimal, 
before  its  root  can  be  found. 

16.  What  is  the  square  root  of  |  ?  Ans.  .53-f-. 

17.  What  is  the  square  root  of  ^fy  ? 

18.  What  is  the  square  root  of  y\  ? 

19.  What  is  the  square  root  of  7|? 

CUBE    ROOT    OF    NUMBERS. 
140.    The  Cube  Root  of  a  number  is  a  number  which, 
taken    three   times  as    a    factor,  will    produce    the   given 
number. 

141*    TJie  cube  of  a  number  consists  of  three   times   as 

many  figures  as  the  root,  or  of  one  or  two  less  than  three 
times  as  many. 

Roots,           1,                    10,                    100,  1000. 

Cubes,           1,                 1000,            1000000,  1000000000. 

The  cube  of  any  number  less  than  10  must  be  less  than  1000; 
but  any  number  less  than   10  is  expressed  by  one  figure,  and  any 


EVOLUTION.  143 

number  less  than  1000  by  less  than  four  figures;  i.  e.  the  cube  of 
a  number  consisting  of  one  figure  is  a  number  of  less  than  four 
figures.  The  cube  of  any  number  between  10  and  100  must  be  be- 
tween 1000  and  1000000;  i.  e.  must  contain  more  than  three  figures 
and  less  than  seven.  And  in  the  same  way  we  see  that  the  cube 
of  any  number  between  100  and  1000  must  contain  more  than  six 
figures  and  less  than  ten. 

Hence,  to  ascertain  the  number  of  figures  in  the  cube 
root  of  a  given  number, 

Beginning  at  units,  point  off  the  number  into  periods  of 
three  figures  each ;  there  will  he  as  many  figures  in  the  root 
as  there  are  periods,  and  for  the  incomplete  period  at  the 
left,  if  any,  one  moi^e. 

142.    To  extract  the  cube  root  of  a  number. 

1.    Find  the  cube  root  of  428T5. 

From  the  preceding  explanation,  it  is  evident  that  the  cube  root 

of  42875  is   a  number  of  two  figures,  and   that   the   tens  figure   of 

the  root  is  the  cube  root  of  the  greatest  perfect  cube   in  42  ;  i.    e. 

^I~21,  or  3.     Now,  if  we  represent  the  tens  of  the  root  by  a  and  the 

units   by  6,  a  -\-h  will   represent  the   root,  and   the  given  number 

will  be 

(a  -I-  &)»  =  rt«  -f  3  0==  &  +  3  a  6^  _|_  ^,3, 

Now  0^=  30^  =  27000; 

therefore,     Z  a^h  -\-  Z  aV  -\-  h^  =  A2%lb  —  27000  =  15875. 
But  3a=&-f  3ai2_^6^=  (3a2+ 3rt6  +  62)6. 

If  therefore  15875  is  divided  by  3  a^  _[_  3  «  &  _|_  ft2  \^  ^'^\\  gj^^  j^ 
the  units  of  the  root.  But  &,  and  therefore  3  a  6  -j-  W,  a  part  of 
the  divisor,  is  unknown,  and  we  must  use  3a^=^2700  as  a  trial 
divisor.  15875 -|- 2700,  or  158-^27  =  5,  a  number  that  cannot 
be  too  small  but  may  be  too  great,  because  we  have  divided  by  3  a^ 
instead  of  ihe  true  divisor,  Z  a^  -\-  Z  ah  -\- V^.  Then  &  =  5,  and 
3a2_j_  3a^_|_j2^  2700  -f-  450  -}-  25  =  3175,  the  true  divisor; 
and  (3  a'*  -f-  3  a  6  -|-  6^)  6  =  3175  X  5  =  15875,  and  therefore  5  is 
the  unit's  figure  of  the  root,  and  35  is  the  required  root.  The  work 
will  appear  as  follows:  — 


144  ELEMENTARY  ALGEBRA. 

OPERATION. 

4  2  8  T  5  (35  Root. 
27 
Trial  divisor,  3  a^  =  2  7  0  0 

Sab—     45  0 
^==        25 


True  divisor,  S  a^ -\- Sab  +  b^  —  3  11  b 


15  8  7  5  Dividend. 


15875 


Hence,  to  extract  the  cube  root  of  a  number, 

RULE. 

Separate  the  number  into  periods  of  three  figures  each,  by 
placing  a  dot  over  units,  thousands,  &c. 

Find  the  greatest  cube  in  the  left-hand  period,  and  place 
its  root  at  the  right. 

Subtract  this  cube  from  the  left-hand  period,  and  to  the  re- 
mainder annex  the  next  period  for  a  dividend. 

Square  the  root  figure,  annex  two  ciphers,  and  multiply 
this  result  by  three  for  a  trial  divisor  ;  divide  the  dividend 
by  the  trial  divisor,  and  place  the  quotient  as  the  next  figure 
of  the  root. 

Multiply  this  root  figure  by  the  part  of  the  root  previously 
obtained,  annex  one  cipher  and  multiply  this  result  by  three ; 
add  the  last  product  and  the  square  of  the  last  root  figure  to 
the  trial  divisor,  and  the  sum  will  be  the  true  divisor. 

Multiply  the  true  divisor  by  the  last  root  figure,  subtract 
the  product  from  the  dividend,  and  to  the  remainder  annex 
the  next  period  for  a  dividend. 

Find  a  new  trial  divisor,  and  proceed  as  before,  until  all 
the  periods  have  been  employed. 

Note  1.  —  The  notes  under  the  rule  in  square  root  (Art  139) 
apply  also  to  the  extraction  of  the  cube  root,  except  that  00  must 
be  annexed  to  the  trial  divisor  when  the  root  figure  is  0,  and  after 
placing  the  first  point  over  units  the  point  must  be  placed  over 
every  third  figure  from  this. 

Note  2.  —  As  the  trial  divisor  may  be  much  less  than  the  true 


EVOLUTION. 


145 


divisor,   the   quotient  is   frequently  too   great,  and    a    less  number 
must  be  placed  in  the  root. 

2.   Find  the  cube  root  of  18191447. 


1st  Trial  Divisor, 


OPERATION. 


3  a''    =1200 

3a6  =     360 

6*^  =        3  6 


18191447(263 

8 

10191        1st  Dividend. 


1st  True  Divisor,  3  a^ -{- 3  ah -\- b^  =15  9  6]     9576 
2d  Trial  Divisor,  Ba^    =20  2  80  01 

Sab  ==        2340 
6^=  9 


6  1544  7     2d  Div. 


615447 


2d  True  Divisor,  Sa^-fSaft  +  ft^  =205149     

We  suppose  at  first  that  a  represents  the  hundreds  of  the  root 
and  6  the  tens:  proceeding  as  in  Ex.  1,  we  have  26  in  the  root 
Then  letting  a  represent  the  hundreds  and  tens  together,  i.  e.  26 
tens,  and  b  the  units,  we  have  3  a^  the  2d  trial  divisor,  =  202800 ; 
and  therefore  6  =  3;  and  3  a^  -{-  3  a  b  -\- b\  the  2d  true  divisor, 
=  205149;  and  263  is  the  required  root. 

Note.  —  Though  the  1st  trial  divisor  is  contained  more  than  8 
times  in  the  dividend,  yet  the  root  figure  is  only  6. 

3.    Find  the  cube  root  of  687I6.4T. 

OPERATION. 


6  8  7  1  6.4  T  0  (4  0.9  5+ 
64 


4  8  0  0.0  0 

1  0  8.0  0 

.8  1 


4  7  1  6.4  T  0 


49  0  8.8  1 

60  18.43  0  0 

6.13  50 

.0  0  25 

6  0  24.56  7  5 


4417.929 
2  9  8.541000 


25^L2  2  8_37^ 
4  7.312625 


146  ELEMENTARY   ALGEBRA. 

4.  Find  the  cube  root  of  2924207.  Ans.  143. 

5.  Find  the  cube  root  of  8120601.  Ans.  201. 

6.  Find  the  cube  root  of  36926037. 

7.  Find  the  cube  root  of  67917.312. 

8.  Find  the  cube  root  of  46417.8. 

9.  Find  the  cube  root  of  .8.  Ans.  .928+. 

10.  Find  the  cube  root  of  .17164. 

11.  Find  the  cube  root  of  .0064. 

12.  Find  the  cube  root  of  25.00017. 

13.  Find  the  cube  root  of  2.7. 

Note.  —  As  a  fraction  is  involved  by  involving  both  numerator  and 
denominator  (Art.  127),  the  cube  root  of  a  fraction  is  the  cube  root 
of  the  numerator  divided  by  the  cube  root  of  the  denominator. 

14.  What  is  the  cube  root  jV  ?  ^^s-  f  • 

15.  What  is  the  cube  root  of  j^^  ? 

16.  What  is  the  cube  root  of  f^f  ?     ^  =  ^f|. 

Ans.  ^. 

Note.  —  If  both  terms  of  the  fraction  are  not  perfect  cubes,  and 
cannot  be  made  so,  reduce  the  fraction  to  a  decimal,  and  then  find 
the  cube  root  of  the  decimal.  A  mixed  number  must  be  reduced 
to  an  improper  fraction,  or  the  fractional  part  to  a  decimal,  be- 
fore its  root  can  be  found. 

17.  What  is  the  cube  root  of  ^^  ?  Ans.  .899-f-. 

18.  What  is  the  cube  root  of  ^\  ? 

19.  What  is  the  cube  root  of  3f  ? 

20.  What  is  the  cube  root  of  117^1' 


EVOLUTION.  147 


EVOLUTION    OF    MONOMIALS. 

143.  As  Evolution  is  the  reverse  of  Involution,  and 
since  to  involve  a  monomial  (Art.  126)  we  multiply  the 
exponent  of  each  letter  by  the  index  of  the  required 
power,  and  prefix  the  required  power  of  the  numerical 
coefficient, 

Hence,  to  find  the  root  of  a  monomial, 

RULE. 
Divide  the  exponent  of  each  letter  by  the  index  of  the  re- 
quired root,   and  prefix  the  required  root  of  the  7iumerical 
coefficient. 

Note  1.  —  The  rule  for  the  signs  is  given  in  Art.  136.  As  an 
even  root  of  a  positive  quantity  may  be  either  positive  or  negative^ 
we  prefix  to  such  a  root  the  sign  ±  ;  read,  plus  or  minus. 

Note  2.  —  It  follows  from  this  rule  that  the  root  of  the  product 
of  several  factors  is  equal  to  the  product  of  the  roots.  Thus, 
y/"36  =  v^l  y/'O  =  6. 

1.  Find  the  cube  root  of  8a^y^  Ans.  2xy^. 

2.  Find  the  square  root  of  4  x^.  Ans.   ±2x. 

3.  Find  the  third  root  of  —  125  a^x. 

Ans.  —  5  a^  x^. 

4.  Find  the  fourth  root  of  81  a"^  b. 

Ans.    ±  3  a-i  h^. 
6.    Find  the  fifth  root  of  S2  a^H^  Ans.  2a^b^. 

6.  Find  the  cube  root  of  —  129  x^  if. 

Ans.  —  9xy*. 

7.  Find  the  fourth  root  of  256  a;'*/. 

8.  Find  the  cube  root  of  —  512  a-^^ 

9.  Find  the  fifth  root  of  243  ar^/^ 


148  ELEMENTARY  ALGEBRA. 

Note.  —  As  a  fraction  is  involved  by  involving  both  numerator 
and  denominator  (Art.  127),  a  fraction  must  be  evolved  by  evolving 
both  numerator  and  denominator. 

4  ^2  2  a 

10.   Find  the  square  root  of  — j-  Ans.  ±  r--^- 

Perform  the  operations  indicated  in  the  following  ex- 
pressions :  — 


11.  v^—T29  aH«c». 

12.  (49a2a;4/)i 

13. 


y  36iV 


14.  ^a'^af*". 

15.  (266a*a:i<^3^i<5)^. 


16.    ^81aH^ 


IT.   \^a'^b^"'(f\ 


SQUARE    ROOT    OF    POLYNOMIALS. 

144*  In  order  to  discover  a  method  for  extracting  the 
square  root  of  a  polynomial,  we  will  consider  the  rela- 
tion of  a  +  ^  to  its  square,  a^  -{-  2  a  h  -\-  b^.  The  first 
term  of  the  square  contains  the  square  of  the  first  term 
of  the  root ;  therefore  the  square  root  of  the  first  term  of 
the  square  will  be  the  first  term  of  the  root.  The  second 
term  of  the  square  contains  twice  the  product  of  the  two 
terms  of  the  root ;  therefore,  if  the  second  term  of  the 
square,  2  a  b,  is  divided  by  twice  the  first  term  of  the 
root,  2  a,  we  shall  have  the  second  term  of  the  root  b. 
Now,  2ab-{-  b'=  (2  a-\-b)  b;  therefore,  if  to  the  trial 
divisor  2  a  wc  add  b,  when  it  has  been  found,  and  then 


EVOLUTION.  149 

multiply  the  corrected  divisor  by  h,  the  product  will  be 
equal  to  the  remaining  terms  of  the  power  after  a^  has 
been  subtracted. 

The  process  will  appear  as  follows :  — 

OPERATION.  Having  written  a,  the  square 

a^  -\-  2  a  h  -\-  h^  {a  -\-  b         root  of  aS  in  the  root,  we  sub- 

a^  tract  its  square   («^)   from    the 

2a-\-b)2ab-\-h'^  g^^^"     polynomial,     and     have 

2  ab  A-  b^  2  a  6  -|-  ^'^  left.      Dividing  the 

first    term    of    this    remainder, 

2  a  &,  by  2  a,  which  is  double  the  term  of  the  root  already  found, 
we  obtain  6,  the  second  term  of  the  root,  which  we  add  both  to 
the  root  and  to  the  divisor.  If  the  product  of  this  corrected  divisor 
and  the  last  term^  of  the  root  is  subtracted  from  2ah  -\-h^,  nothing 
remains. 

145.  Since  a  polynomial  can  always  be  written  and 
involved  like  a  binomial,  as  shown  in  Art.  131,  we  can 
apply  the  process  explained  in  the  preceding  Article  to 
finding  the  root,  when  this  root  consists  of  any  number 
of  terms. 

1.  Find  the  square  root  of  0^+  2ab-\-  b'^  —  2ac—2bc  +  c\ 

OPERATION. 

^.2 


2a  +  b)2ab  +  b'^ 
2ah  +  h'^ 


2a-\-2b~c)—2ac  —  2bc-\-c^ 
—  2ac  —  2bc-{-c'^ 

Proceeding  as  before,  we  find  the  first  two  terms  of  the  root  a-\~b. 
Considering  a  -\~  b  as  a  single  quantity,  we  divide  the  remainder 
—  2  ac  —  2bc  -\-  c^  by  twice  this  root,  and  obtain  —  c,  which  we 
write  both  in  the  root  and  in  the  divisor.  If  this  corrected  divisor 
is  multiphed  by  —  c,  and  the  product  subtracted  from  the  dividend, 
nothing  remains. 


150  ELEMENTARY  ALGEBRA. 

Hence,  to  extract  the  square  root  of  a  polynomial, 

RULE. 

Arrange  the  terms  according  to  the  powers  of  some  letter. 

Find  the  square  root  of  the  first  term,  and  write  it  a^  the 
first  term  of  the  root,  and  subtract  its  square  from  the  given 
polynomial. 

Divide  the  remainder  by  double  the  root  already  found, 
and  annex  the  result  both  to  the  root  and  to  live  divisor. 

Multiply  the  corrected  divisor  by  this  last  term  of  the  root, 
and  subtract  the  product  from  the  last  remainder.  Proceed 
as  before  with  the  remainder,  if  there  is  any. 

2.  Find  the  square  root  of  4  x^  —  4:Xi/^  -\-  ^*. 

Ans.  2x  —  y^. 

3.  Find   the   square   root   ofa^  -\-2ab-\-b'^-\-4:ac 
+  4  5  c  +  4  c*.  Ans.  a-\-  b-\-2c. 

4.  Find  the  square  root  of  9x*  —  I2a^ -{- 4:X^ -\- 6ax^ 

—  4:ax-\-a^.  Ans.   Sx'^  —  2x-\-a. 

5.  Find  the  square  root  of4a^-|~  Sab  —  4:a  -\-  4:b'^ 

—  4i+l  Ans.  2a  +  2b—l. 

6.  Find  the  square  root  of  25  a:*  —  lOx*  -j-  6x^  —  x 
H-i.  Ans.  5x^  —  x+^. 

1.   Find  the  square  root  of  x^  -{-2x^  —  x*  —  2x*  +  ^*- 

8.  Find  the  square  root   of  4  0^  —  4a6  +  b'^  —  iac 

Ans.  2  a  —  b  —  c  —  d. 

9.  Find   the   square   root   of  x^  —  4a:^  +  6  a:*  —  6x* 
-\-5x^  —  2x+  1. 

10.    Find    the    square    root    of   4  a*  +   8«'6  —   Sa'H^ 

—  12a6»  +  9i*. 


EVOLUTION.  151 

Note  l.  —  According  to  the  principles  of  Art.  136,  the  signs  of 
the  answers  given  above  may  all  be  changed,  and  still  be  correct. 

Note  2.  —  No  binomial  can  be  a  perfect  square.  For  the  square 
of  a  monomial  is  a  monomial,  and  the  square  of  the  polynomial  with 
the  least  number  of  terms,  that  is,  of  a  binomial,  is  a  trinomial. 

Note  3.  —  A  trinomial  is  a  perfect  square  when  two  of  its  terras 
are  perfect  squares  and  the  remaining  term  is  equal  to  twice  the 
product  of  their  square  roots.     For, 

(a-\-by==a^-\-2ab-{-W 
(a  —  hf  =  a^  —  2ab  -\-  I? 

Therefore  the  square  root  of  a^  ±  2  a  6  -]-  6^  is  a  ±  6.     Hence,  to 
obtain  the  square  root  of  a  trinomial  which  is  a  perfect  square, 

Omitting  the  term  that  is  equal  to  twice  the  product  of  the  square 
roots  of  the  other  tivo,  connect  the  square  roots  of  the  other  two  by  the 
sign  of  the  term  omitted. 


5; y 

2         2 


11.  Find  the  square  root  of  ~  —  ^  +  ~- 

Ans 

12.  Find  the  square  root  oi  x^  -\-2x-\-  1. 

Ans.  X  -\-\. 

13.  Find  the  square  root  of  43:^ —  ^xy  -\-^y^. 
.  14.    Find  the  square  root  of  -  —  2  a  i  +  9  5^ 

15.  Find  the  square  root  of  16/ _|_  40^2^2  _|_  25^4 

Note.  —  By  the  rule  for  extracting  the  square  root,  any  root  whose 
index  is  any  power  of  2  can  be  obtained  by  successive  extractions 
of  the  square  root.  Thus,  the  fourth  root  is  the  square  root  of  the 
square  root ;  the  eighth  root  is  the  square  root  of  the  square  root  of 
the  square  root;  and  so  on. 

16.  Find  the  fourth  root  of  a^  —  12a«&  +  540^^^ 
—  108  a^i^  4-81  5^  Ans.  a2-^3i. 


152  ELEMENTARY  ALGEBRA. 

11.   Find  the  fourth  root  of  -,  +  ~  +  ::^  +  — ,  4-  1 

X    '    y 

18.    Find  the  fourth  root  of  a:^  —  4x'^  +  ^Ox^  —  IQx^ 
4-19ar*— 16a:»+ lOar^  — 4ar+ 1.       Ans.  ar^  — x+1. 


146.   To  find  any  root  of  a  polynomial. 

Since,  according  to  the  Binomial  Theorem,  when  the  terms  of  a 
power  are  arranged  according  to  the  power  of  some  letter  begin- 
ning with  its  highest  power,  the  first  term  contains  the  first  term 
of  the  root  raised  to  the  given  power,  therefore,  if  we  take  the  re- 
quired root  of  the  first  terra,  we  shall  have  the  first  term  of  the  root. 
And  since  the  second  term  of  the  power  contains  the  second  terra  of 
the  root  raidtiplied  by  the  next  inferior  power  of  the  first  term  of  the 
root  with  a  coefficient  equal  to  the  index  of  the  root,  therefore  if  we 
divide  the  second  term  of  the  power  by  the  first  term  of  the  root  raised 
to  the  next  inferior  power  with  a  coefficient  equal  to  the  index  of  the 
root,  we  shall  have  the  second  term  of  the  root.  In  accordance  with 
these  principles,  to  find  any  root  of  a  polynomial  we  have  the  foUowing 

RULE. 

Arrange  the  terms  according  to  the  powers  of  some  letter. 

Find  the  required  root  of  the  first  term,  and  vjrite  it  as 
(he  first  term  of  the  root. 

Divide  the  second  term  of  the  polynomial  by  the  first  term 
of  the  root  raised  to  the  next  inferior  power  and  multiplied 
by  the  index  of  the  root. 

Involve  the  whole  of  the  root  thus  found  to  the  given  power, 
and  subtract  it  from  the  polynomial. 

If  there  is  any  remainder,  divide  its  first  term  by  the  di- 
visor first  found,  and  (lie  quotient  will  he  the  third  term  of 
the  root. 

Proceed  in  this  manner  till  the  power  obtained  by  involv- 
ing the  root  is  equal  to  the  given  polynomial. 


EVOLUTION.  153 

Note  1.  —  This  rule  verifies  itself.  For  the  root,  whenever  a  new 
term  is  added  to  it,  is  involved  to  the  given  power,  and  whenever  the 
root  thus  involved  is  equal  to  the  given  polynomial,  it  is  evident  that 
the  required  root  is  found. 

Note  2.  —  As  powers  and  roots  are  correlative  words,  we  have 
used  the  phrase  given  power,  meaning  the  power  whose  index  is  equal 
to  the  index  of  the  required  root,  and  the  phrase  next  inferior  power 
meaning  that  power  whose  index  is  one  less  than  the  index  of  the 
required  root. 

1.    Find  the  cube  root  of  a^  — 3  a5  + 5«^  — 3a— 1. 

OPERATION. 

Constant  divisor,  3  a*)  a^  —  3  a^  +  5  a^  —  3  a  —  1  (a^  —  a  —  1 

—  3  a*,  1st  term  of  remainder. 


a«  __  3  a^  +  5  a^  —  3  a  —  1 

The  first  term  of  the  root  is  a^,  the  cube  root  of  a^.  a^  raised 
to  the  next  inferior  power,  i.  e.  to  the  second  power,  with  the  co- 
efficient 3,  the  index  of  the  root,  gives  3  a*,  which  is  the  constant 
divisor.     —  3  a^,   the  second  term  of  the   polynomial,    divided   by 

3  a*,  gives  —  a,  the  second  term  of  the  root,  {d?  —  ay  =  a^  —  3  a* 
-j-  3  a*  —  a' ;  and  subtracting  this  from  the  polynomial,  we  have  —  3  a* 
as  the  first  term  of  the  remainder.     —  3  a*  divided  by  3  a*  gives 

—  1,  the  third  term  of  the  root,  (a^  —  a — 1)^=  the  given  poly- 
nomial, and  therefore  the  correct  root  has  been  found. 

2.  Find  the   fourth   root   of  16  a;^  —  32  x^  /  +  24  a;^  ^/^ 

OPERATION. 

4  X  (2a;)'  =  32r^)  16  a;*  —  32  a:^  y^  -[-  24  r^  ?/*  —  8  ar  /  -f  ?/  (2  x  —  ?f 

16x*  — 32arV-f  24:  sc"  y^  —  S  x  y" -f- f 

3.  Find  the  cube  root  of  a^  +  S  aH +  Z  ab^  +  b^ —  Sa'^  c 

—  6abc  —  Sb^c  +  Sac''-\-Sbc''  —  c\ 

4.  Find  the  fourth  root  of  16  a'  c'  —32a^c^x  +  24  a^  c^  x" 

7* 


154  ELEMENTARY   ALGEBRA. 


SECTION    XVII. 

RADICALS. 
147.    A  Radical  is  the  indicated  root  of  any  quantity, 
as  \/x,    a^,    \/"2,    3^,   &c. 

,      148.    In  distinction    from  radicals,  other   quantities   are 
called  rational  quantities. 

149.  The  factor  standing  before  the  radical  is  the  co- 
efficient of  the  radical.  Thus,  2  is  the  coeflScient  of  \/2 
in  the  expression  2\/2. 

150.  Similar  Radicals  are  those  which  have  the  same 
quantity  under  the  same  radical  sign.  Thus,  \/a,  2  \/a, 
and  X  a/ a  are  similar  radicals  ;  but  2  \/«  and  2  \/6,  or 
2x^  and  2x"  are  dissimilar  radicals. 

151.  A  Surd  is  a  quantity  whose  indicated  root  cannot 
be  found.     Thus,  \/2  is  a  surd. 

The  various  operations  in  radicals  are  presented  under 
the  following  cases. 

CASE  I. 

152.  To  reduce  a  radical  to  its  simplest  form. 

NoTB.  —  A  radical  is  in  its  simplest  form  when  it  contains  no 
factor  whose  indicated  root  can  be  found. 

1.    Reduce  ^x/TSa^'J  to  its  simplest  form. 

OPERATION. 


We  first  resolve   75  a'  h  into  two  factors,  one  of  which,  25  a",  is 
the  greatest  perfect  square  which  it  contains ;  then,  as  the  root  of 


RADICALS.  155 

the  product  is  equal  to  the  product  of  the  roots  (Art.  143,  Note  2), 
we  extract  the  square  root  of  the  perfect  square  25  a^,  and  annex 
to  this  root  the  factor  remaining  under  the  radical.     Hence, 

RULE. 
Resolve  the  quantity  under  the  radical  sign  into  two  fac- 
tors, one  of  which  is  the  greatest  perfect  power  of  the  same 
name  as  the  root.  Extract  the  root  of  the  perfect  power, 
multiply  it  by  the  coefficient  of  the  radical,  if  it  has  any,  and 
annex  to  the  result  the  other  factor,  with  the  radical  sign 
between  them. 

Reduce    the    following    expressions    to    their    simplest 
form  :  — 


2.  -v/12a:.  Ans.  2  ^'6x. 

3.  s^4:9x\  Ans.  1  x^  sTx. 


4.    ^12an\  Ans.  2a4^9b^. 


5.    5-^64a6*.  Ans.   I0b^4=a. 


6.    S\^U1aH\  Ans.  2lab^s/S. 

1.    25  ^~56x.  Ans.  50  ^Tx. 


8.   4.\/l2Sx^y. 


9.    -^343x». 


Y    i28V 


10-/ -W 


/   27  a" c  __     /9  a'      /?_£__  ?^     / 
Y    128^  ~y    64^  Y    2y  ~8a:^^  Y 


3c  ■ 

—  I    Ans. 
2y 


11.  ^ 


12.   \/16x2y2_32^4^6 


^  16  xV  _  32  x\y^  =:  V'  16  a:^/  V"  1  —  2  ipV 
=  4xy  Vl  —  2x^^,  Ans. 


15G  ELEMENTARY  ALGEBRA. 


13.    4:\/Sla^c  +  21a^  Ans.   12a^3c  +  l. 


14.    (a  +  i)  V3a2_6a6-[-362.    Ans.  (a^  _  ^)  >^  3. 


15.  •?  ^250a:«/— 125arV. 

16.  (x  —  1/)  {a-'x  —  a'^y)^. 
IT.    (a«+  a«Z;2^i 


18. 

V- 

-16. 

16 

19. 

^- 

1250. 

Vl6  V— 1=4\/-^,  Ans. 


20.    Vl9a'  — 4^. 


153.  When  a  fraction  is  under  the  radical  sign,  it  can 
be  transformed  so  as  to  have  only  an  integral  quantity 
under  the  radical  sign,  by  multiplying  both  terms  of  the 
fraction  by  (hat  quantity  which  will  make  its  denominator  a 
perfect  power  of  the  same  name  as  the  root,  and  then  re- 
moving a  factor  according  to  the  Bute  in  Art.  152. 


1.   Reduce  yy  -   to  its  simplest  form. 


OPERATION. 


(/5  =  ^/^  =  V^^^^=5^ 

Transform  each  of  the   following  expressions  so  as  to 
have  only  an  integral  quantity  under  the  radical  sign. 

2.   l^l-  Ads.  ^  >/ 6. 

^|.  Ans.  VV9. 

l\/l-  ■  Ans.  1^343. 

-     a      I   17  ^  1       , 


3.    4 
4. 


RADICALS.  157 


6.   ^^^  .       A„S.1V^. 


10 


833-  Ans.  g-5^/30. 


-•'^/^ 


^■2^8-.-  Ans.  -VUx. 

^    —  _ 

11.    («  +  ^)  y/^.  Ads.  V^^^=^. 

CASE    II. 
154*   To   reduce   a  rational   quantity  to   the  form   of  a 
radical. 

1.  Reduce  Sx^  to  the  form  of  the  cube  root. 

OPERATION.  Since  3  a;''  is  to  be  placed 

3  <^2  ^    3^27  x^  under  the  form  of  the  cube 

root  without  changing  its 
value,  we  cube  it  and  then  place  the  radical  sign,  ^,  over  it.  It  is 
evident  that  (/"27^  =3  3^.     Hence, 

RULE. 
Involve  the  quantity  to  the  power  denoted  by  the  index  of 
the  root  required,  and  place  the  corresponding  radical  sign 
over  the  power  thus  produced. 

2.  Reduce  4a^5  to  the  form  of  the  square  root. 


Ans.  \^l6aH\ 
3.    Reduce  2aPc~'^  to  the  form  of  the  fifth  root. 


1   4 


Ans.  4/S2a'b^'c-^. 


4.   Reduce  -  a^  c^  to  the  form  of  the  cube  root. 


158  ELEMENTARY   ALGEBRA. 

2  cPh 
6.    Reduce  -^ — :  to  the  form  of  the  fourth  root. 
Zxyk 

6.    Reduce  x  —  2y  to  the  form  of  the  square  root. 

Ads.  \/ x^  —  4a:y-|-4y*. 

155t  On  the  same  principle  the  rational  coeflBcient  of  a 
radical  can  be  placed  under  the  radical  sign,  by  involv- 
ing the  coefficient  to  a  power  of  the  same  name  as  the  root 
indicated  by  the  i^adical  sign,  multiplying  it  by  the  radical 
quantity,  and  placing  the  given  radical  sign  over  the  product. 

1.  Place  the  coefficient  of  ht^ly  under  the  radical 
sign. 

OPERATION. 


6  >^  2^  =  /^  125  ><^  2y  =  -^  250y 

In  the  following  examples,  place  the   coefficient   under 
the  radical  sign. 


2.    Z4/^x^y.  Ans.  -</ 324 a:»y. 


3.    2xy4^2x'y.  Ans.  -^IGar^y*. 

5.    ^v/li. 


6.  (a_-i)^_^.  Ans.  ^ aJ" —  2d'b -^  ab\ 

7.  4.xys/l  —  2a:2y*. 

CASE    III. 

156.    To   reduce    radicals    having    different    indices    to 
equivalent  ones  having  a  common  index. 

1.    Reduce  \/  a  and  f^  b  io   equivalent   radicals   having 
a  common  index. 


RADICALS.  159 

OPERATION.  In  this  case  we  write  the  radicals 

i  I  6/ — k  with    their    fractional    indices ;    and 

.,  _  then,  as  the  denominator  is  the  in- 

b^  =  P  =  ^  P  dex  of  the   root,  in   order  that  the 

two  radicals  may  have  the  same 
root-index,  we  reduce  the  fractional  indices  to  equivalent  ones  hav- 
ing a  common  denominatoro  It  is  evident  that  we  have  not  changed 
the  values  of  the  given  radicals  by  the  process.     Hence, 

RULE. 

Reduce  the  fractional  indices  to  equivalent  ones  having  a 
common  denominator ;  involve  each  quantity  to  the  power  de- 
noted by  the  numerator  of  the  reduced  index,  and  indicate 
the  root  denoted  by  the  denominator. 

2.  Reduce  \/ 2  and  \/ 3  to  equivalent  radicals  having 
a  common  index. 


3^  =  3^'^  =  >^F  =  ^  2Y  ) 


Ans. 


3.  Reduce  \/  ^  and  v^  ^  to  equivalent   radicals   having 
a  common  index.  Ans.  ^^ib  ^"^  ^  s^- 

4.  Reduce   i  /  -  and   tV  -  to  equivalent  radicals  having 
a  common  index. 


5.  Reduce  \/  a,  a^  a  —  b,  and  \/  a  -\-  b  to    equivalent 
radicals  having  a  common  index. 

Ans.  i^^^  ^  (a^^y,  and  ^  (a+~by. 

6.  Reduce  \/  2,    -C^  4,   and  -^  3   to   equivalent  radicals 
having  a  common  index. 

7.  Reduce  \/ x  and  \^  y   to   equivalent  radicals   having 
a  common  index.  Ans.   v'a;"*  and  v^y*. 


160  ELEMENTARY  ALGEBRA. 

CASE    IV. 
157.   To  add  radical  quantities. 

1.  Add  tsf  X  and  ^/ y.  Ans.  ^fx -^  ^~y. 
It  is  evident  that  the  addition  can  only  be  expressed. 

2.  Add  Zi^x  and  bsfx.  Ans.  8 /v/x . 

It  is  evident  that  3  times  the  y/  x  and  5  times  the  ^~x  make  8 
times  the  y/  x. 

3.  Add'V'S  and  \/5U  together. 

OPERATION.  In  this  case  we  make  the  radi- 

^8  -_  2  ^~^  cal  parts  similar  by  reducing  them 

-V-- -^  to  their  simplest  form  (Art.   152), 

^        HI — — —  and  then  add  their  coefficients  as 

Sum  =1^/2  in  Example  2.     Hence, 

RULE. 

Make  the  radical  parts  similar  when  they  are  not,  and 
prefix  the  sum  of  the  coefficients  to  the  common  radical.  If 
the  radical  parts  are  not  and  cannot  he  made  similar^  corir 
nect  the  quantities  with  their  proper  signs. 


4.    Add  2^^0ax  and  S\/9Sax.     Ans.  31\/2aa:. 


5.  Add  4-^24^8  and  a:/^81.  Ans.  llx-^3. 

6.  Add  \/27  and  \/363.  Ans.   14  \/ 3. 


7.  Add  -^512a:<  and  Ayi62y*. 

Ans.  (4:X  +  Si/)^2. 

8.  Add  A^5  and  \/"^. 

Vi  =  \/2»F\^5  =  iV5;   V5  +  iV5  =  fV5,  Ans 

9.  Add  t/^  and  ^1|^.  Ans.  -g^  ^T2. 


KADICALS.  361 

10.  Add  \/|,  10\/^V^  and  6a/~20.         Ans.  13  V^- 

11.  Add  V'lO  and  \/^. 

CASE    V. 
158.   To  subtract  one  radical  from  another. 

1.   From  A/lb  take  ^/~21. 

OPERATION.  We  make  the  radical  parts  sim- 

,-hT r     /~o  il^r  hy  reducing  them  to  their  sim- 

plest form  (Art.  152).  And  3  y/^ 
taken  from  5  y/  3  evidently  leaves 
2  v/"3.     Hence, 


V27  i=3v/3 
2^/3 


EULE. 
Make  (he  radical  parts  similar  when  they  are  not,  sub- 
tract the  coefficient  of  the  subtrahend  from  that  of  the  min- 
uend, and  prefix  the  difference  to  the  common  radical.  If 
the  radical  parts  are  not  and  cannot  be  made  similar,  indi- 
cate the  subtraction  by  connecting  them  with  the  proper  sign. 

2.    From  /^"sT  take  /^  3.  Ans.  2  a^J. 


3.    From  9\^a^xy^  take  Za^/xy^. 


Ans.   ^ays/x. 


4.  From  T  \/  20  ar  take  4  \/  45  a:.  Ans.  2  >/  ^  ^. 

6.  From  -^500  take  -^l08.  Ans.  2  ^4. 

6.  From  2  \/7^  take  ^/^.  Ans.  t^^  V^. 

T.  From  \/  f  take  \/^.  Ans.  -i^  \/Ta 


8.    From  2/^n6a:«  take  >^891a;S. 


9.    From  a  /^a:^  take  1  ^  a^x". 


10.   From  -^1174  take  -^1892. 


162  ELEMENTARY    ALGEBRA. 

CASE   VI. 
159.    To  multiply  radicals. 

1.    Multiply  3/v/«  by  h  m/I). 

OPERATION. 

3  \/ a  X  5  s/1)  =  3  X  5  X  \/ «  X  \/"^  =15  's/ah 

As  it  makes  no  difference  in  what  order  the  factors  are  taken, 
we  unite  in  one  product  the  numerical  coefficients ;  and  ^  a  y^  ^  h 
=  ^ab  (Art.  143,  Note  2). 


2.    Multiply  4\/2a6  by  5\/Say. 


OPERATION. 


We     reduce     the    radical 

parts  to    equivalent    radicals 

4vza6r=    4\/    8a  6  having     a     common      index 

5^Sax=    5^    9a^x^  (Art.  156),  and  then  multi- 

Product  =  20  ^l2aW^^  P^>^  ^  ^"  *^^  preceding  ex- 

ample. 

S.  Multiply  >^  a  hy  \f  a. 

OPERATION. 

,       a^  X  a^  =  ^cFX~c^—\f~i^\  or  J 

Multiplying  as  in  the  preceding  examples,  we  have  ^  a*,  or  a*  ; 
but  t  =  ^  +  ^ ;  i-  e.  the  Index  of  the  product  is  the  sum  of  the 
indices  of  the  factors. 

From  these  examples  we  deduce  the  following 

RULE. 

I.  Reduce  the  radical  parts,  if  necessary,  to  equivalent 
radicals  having  a  common  index,  and  to  the  product  of  the 
radical  parts  placed  under  the  common  radical  sign  prefix 
the  product  of  their  coefficients. 

II.  Boots  of  the  same  quantity  are  multiplied  together  by 
adding  their  fractional  indices. 


RADICALS. 

163 

4. 

Multiply  3  \/ 10  by  4  V  5. 

Ans.  60  V  2. 

6. 

Multiply  ^f^ax^  by  a\/a:. 

Ans.  ^axt^d^x. 

6. 

Multiply  av'c  by  hf^c. 

Ans.  a  6  c. 

T. 

Multiply  isf  X  by  i^  x. 
Multiply  ^7  by  V^. 

Ans.  Q^. 

5 

8. 

Ans.  76,  or -^16807. 

9. 

Multiply  s/  ^  bj^  s/  X. 

Ans.  V^'"  +  ". 

10.  Multiply  2  ^/a  +  6  by  6  a:  ^a  +  6. 

Ans.   \2x  ^  {a  +  hy. 

11.  Multiply  \/a?,  Va:,  and  ^a:  together.      Ans.  x^^, 

12.  Multiply  3  V^  by  2  VS.  Ans.  6  ^2. 


13.  Multiply  a^x~^y  by  b\/xy.  Ans.  a^y. 

14.  Multiply  (a +  6)*  by  (a  — 5)^     Ans.  {a^  —  h^)^. 

15.  Multiply  ^Vf  by  3\/7. 

16.  Multiply  2  V'S  by  4  V^. 

CASE  VII. 
160.   To  divide  radicals. 


1.   Divide  60\/15x  by  4V5ar. 

OPERATION.  As  division  is   finding 

60  s^lbx  H-  4  ^/bx=^  15/^/3"  a  quotient  which,  multi- 

plied by  the  divisor,  will 
produce  the  dividend,  the  coefficient  of  the  quotient  must  be  a 
number  which,  multiplied  by  4,  will  give  60,  the  coefficient  of  the 
dividend,  i.  e.  15;  and  the  radical  part  of  the  quotient  must  be  a 
quantity  which,  multiplied  by  y'Sa;,  will  give  v/15a:,  i.e.  ^  Z\  the 
quotient  required,  therefore,  is  15  y'  3. 


164  ELEMENTARY  ALGEBRA. 

2.    Divide  6  \/Ty  by  2  >^ 2y. 


OPERATION. 


6  V42/  -^  2  /^2y  =  6  .^64^  -j-  2  >^4y2  =  3  >^  16y 

We  reduce  the  radical  parts  to  equivalent  radicals  having  a  com- 
mon index  (Art.  155),  and  then  divide  as  in  the  preceding  example. 

3.    Divide  /sj  a  by  /^  a. 

OPERATION. 

a*  -f-  a^  r=  ^  a»"^r^2  __  ^-    ^^  „i 

Dividing  as  in  the  preceding  examples,  we  have  ^  a,  or  a«.  But 
\  =  \  —  \\  i-  e.  the  index  of  the  quotient  is  the  index  of  the  divi- 
dend minus  the  index  of  the  divisor. 

From  these  examples  we  deduce  the  following 

RULE. 

I.  Reduce  the  radical  parts,  if  necessary,  to  equivalent 
radicals  having  a  common  index,  and  to  the  quotient  of  the 
radical  parts  placed  under  the  common  radical  sign  prefix 
the  quotient  of  their  coefficients. 

II.  Boots  of  the  same  quantity  are  divided  by  subtracting 
the  fractional  index  of  the  divisor  from  that  of  the  dividend. 

4.    Divide  16  a/ ax  by  8\/a^.  Ans.  2a/~^. 


6.   Divide  4:\/a^  —  l^  hj  2A/a  —  b.  _ 

Ans.  2  \/  «  +  *.- 

6.  Divide  6^2*7  by  3\/3.  Ans.  6 

7.  Divide  \^x  by  v^a;.  Ans.  "v^x*"" 

8.  Divide  ^/"^  by  ^1.  Ans.  r/^ 

9.  Divide  3  by  ^"3.  Ans.  y/S 


RADICALS.  165 

10.  Divide  x  by  ^~x.  Ans.  f^'^, 

11.  Divide  4,ar\^x  by  2a-^/^y,  Ans.  2a^U-' 

12.  Divide  a/T  by  a^T. 

13.  Divide  ^7  by  -^T. 

14.  Divide  /^a  by  -^a! 

15.  Divide?^?  by  I  ^f. 

CASE   VIII. 
161 1   To  involve  radicals. 

1.  Find  the  cube  of  Sa^x, 

OPERATION. 

(3\/^y=  3\/^X  3\/^  X  3\/"ac 

In  accordance  with  the  definition  of  involution,  we  take  the  quan- 
tity three  times  as  a  factor.     By  Art.  159  the  product  is  27y'r*. 

2.  Find  the  square  of  2^  a. 

OPERATION.  ^^  *^^^  ^^^  ^®    ^^^®   "^^^ 

_  X  Z  ^^®  fractional  exponent,  and 

(2  ^  a)2  =  (2  a^y  =  4  a^  found  the  square  of  the  given 

quantity  by  multiplying  its 
exponent  by  the  index  of  the  required  power,  according  to  Art.  126. 
Hence, 

EULE. 

I.  Involve  the  radical  as  if  it  were  rational,  and  placing 
it  under  its  proper  radical  sigh,  prefix  the  required  power 
of  its  coefficient. 

II.  A  radical  can  be  involved  by  multiplying  its  fractional 
exponent  by  tlve  index  of  the  required  power. 


166  ELEMENTARY  ALGEBRA. 

Note.  —  Dividing  the  index  of  the  root  is  the  same  as  multiply- 
ing the  fractional  exponent.  Thus  the  square  o(  ^  a  is  if  a.  For 
(a«)2  =  a^,  or  ^a. 

3.  Find  the  cube  of  3  a?  \/  a. 

Ans.  27  a  x^  V^i  or  27  J  x^. 

4.  Find  the  square  of  4  a^.        Ans.  16  a^,  or  16  ^a^. 

5.  Find  the  fourth  power  of  B\/x.  Ans.  81  x'^. 

6.  Find  the  nth  power  o^uAyx.  Ans.  d^ /^Ixf', 

7.  Find  the  fourth  power  of  5\/i-  Ans.  25. 

8.  Find  the  cube  of  3  \/T.  Ans.  189  VT. 


9.   Find  the  fourth  power  of 
10.   Find  the  cube  of  2V4x. 


CASE  IX. 
162.  To  evolve  radicals. 

1.  Find  the  cube  root  of  8  a^  t^  a^x^. 

OPERATION.  As  the  root  of  the  product  is 

'  ^  ^  (Art.  143,. Note  2),  Ave  prefix  to 

the  cube  root  of  the  radical  part  the  cube  root  of  the  rational  part. 
The  cube  root  of  the  radical  part  must  be  a  quantity  which,  taken 
three  times  as  a  factor,  will  produce  ^a^z';   i.  e.  ^ax, 

2.  Find  the  fourth  root  of  f^  x. 

OPERATION.  In  this  case  we  have  used' 

— —         /  i\i         jL  the  fractional  exponent,  and 

y/  ^a:  =:  \x^)  =  x^,  or  '(/  a;  found  the  fourth  root  by  di- 

viding the  exponent  of  the 
given  quantity  by  the  index  of  the  required  root,  according  to 
Art.  143.     Hence, 


RADICALS.  167 

RULE. 

I.  Evolve  the  radical  as  if  it  were  rational,  and,  placing 
it  under  its  proper  radical  sign,  prefix  the  required  root  of 
its  coefficient. 

II.  A  radical  can  be  evolved  by  dividing  its  fractional 
exponent  by  the  index  of  the  required  root. 

Note.  —  Multiplying  the  index  of  the  root  is  the  same  as  divid- 
ing the  fractional  exponent.  Thus,  the  square  root  of  ^a  is  ^a. 
For  (a3)2   =  a6^  or  ^  a. 

3.  Find  the  square  root  of  ba^^x. 

(5  a  A^Tx)^  z=  (^500^'^)^  =  ^  500  a' X,  Ans. 

4.  Find  the  cube  root  of  a:"^^a^6.  Ans.  i/^r-* 

5.  Find  the  fifth  root  o^x^/s/x.  Ans.  ^/ x. 

6.  Find  the  fourth  root  of  {  ^~J.  Ans.  /^^. 

7.  Find  the  cube  root  of  1  \/3.  Ans.  >^l47. 

8.  Find  the  square  root  of  12\/5. 

POLYNOMIALS   HAVING   RADICAL  TERMS. 

163.  It  appears  from  the  principles  already  established, 
that  the  laws  which  apply  to  calculations  with  quantities 
which  have  exponents,  apply  equally  well  whether  the 
exponents  are  positive  or  negative,  integral  or  fractional. 
The  following  examples,  therefore,  can  be  done  by  rules 
already  given. 

1.    Add  4  a  —  3  \/y  and  3  a  +  2  \/y.    Ans.   la  —  A^i/. 


2.   Add  3  a: +  ^135  and  7:zr  — -^1080.  _ 

Ans.   10  a:  — 3/^5. 


168  ELEMENTARY  ALGEBRA. 

3.  Add  2  V 28  —  V 27  and  2  V 63  +  \/48. 

4.  Subtract  15  a:  —  \/ bO  a  from  13  a:  —  \/^a. 

Ans.  3\/2a  — 2ar. 

5.  Subtract  /s/ aoc^  —  a^ 4:h  from  \/ ax  —  \/i6  3. 

Ans.  /s/ ax  —  x\/ a  —  2\/A. 

6.  Subtract  /i^  32  —  ^"242  from  —  3  -^T  —  7  V  3. 
•?.   Multiply  \/«  —  \/6  by  \/«  —  V^» 

OPERATION. 

\/  a  —  \/  x 


a  —  /s/ ah  —  j>J ax  -\-  s/hx 

8.  Multiply  a;y +  \/a6  by  4 — /^~ah. 

Ans.  4a?y  +  (^  —  ^y)  \/«^  —  «^' 

9.  Multiply  T  +/\/rO  by  6  —  ^10. 

Ans.  32  —  V 10. 

10.  Multiply  \/a-(T-\/6  by  \/a  —  \/6.       Ans.  a  —  h. 

11.  Multiply  VS  — 4/^3  by  V^S  +  \/9. 

12.  Multiply  i  VI  +  T  \/ 3  by  ^  \/l  —  7  \/ 3. 

13.  Divide  t>J ax  -\-  f^ ay  -\-  x  •\- 1>/ xy  by  \/Qf  +  \^a: 

OPEKATION. 

^ ax  4"  ^ 

\/ay  +V^ 


RADICALS.  169 

14.  Divide /\/ac  —  j^^ad  —  ^/hc  -\-  s/^dhj  s/ c — \/rf. 

Ans.  \/ a  —  \/6. 

15.  Divide   (^ x-\-^ x-^cf-y^ -^}^y^-  hj  x^y^. 

16.  Divide  x  —  ^  by  ^Ic  —  ^ y.      Ans.  \/ x -\-  \/y. 
11.    Divide  4:xy-\-4: \/a  h  —  xy  s/ a  h  —  a 6  by  4  —  s/ a b 

18.  Expand  (\/^  +  V7)^-         -^^s.  x  +  2  \/^  +  ^• 

19.  Expand   {a^  —  r^)^.  Ans.  a  _  2  */^  +  i. 

20.  Expand  (V  a  —  V^)*. 

Ans.  a2  — 4a*6i  +  6a6  — 4aU^  +  62. 

21.  Expand    (4  —  v^f .  Ans.    100  —  51^/3. 

22.  Expand  (a-4  —  a;-^)*. 

Ans.  a-t  —  3  a'^  a:-^  +  3  arix'^  —  a?"*. 


23.    Expand  (1-y/iy 


Ans    l-JU-4--^ ^4-i. 

•   16        2v^a        2a        ay^a    "^  a' 

24.  Expand  (^l-y'ly. 

Ans.  ^-^^*  +  xy-i4^  +  f 
4  v^6        '        ^  3v^6      '    9 

25.  Find  the  square  root  of  a  —  2a^6^  -f-  6^. 

Ans.  Va  — a7'6^. 

26.  Find  the  cube  root  of  ic^  —  3  x^^/^  -f-  3  x^/^  —  y. 

Ans.  ic  —  y^. 

2Y.   Find  the  fourth   root   of  16  a  —  32a^/ +  24a*y^ 
—  8  a^i/2  _j_  ^1  Ans.  2  a^  —  t/1 


170  ELEMENTARY  ALGEBRA. 


SECTION   XVIII. 

PURE    EQUATIONS 

WHICH    REQUIRE  IN   THEIR   REDUCTION   EITHER  INVO- 
LUTION OR  EVOLUTION. 

164*  A  Pure  Equation  is  one  that  contains  but  one 
power  of  the  unknown  quantity ;  as, 

\/x  -|-  a  c  =  i,  4  ar^  -j-  3  1=  7,  or  I4.3f  =  ab. 

165.  A  Pure  Quadratic  Equation  is  one  that  contains 
only  the  second  power  of  the  unknown  quantity ;  as, 

6x^—Ua  =  5lb,  af=lScd,  or  ac z'  =  14. 

166(  Radical  Equations,  i.  e.  equations  containing  the 
unknown  quantity  under  the  radical  sign,  require  Invo- 
lution in  their  reduction. 

167.   To  reduce  radical  equations. 

1.    Reduce  \^x  —  3  =  8. 

OPERATION. 

V^  —  3  =      8 
Transposing,  \/ x  =^    11 

Squaring,  x  =  121 


2.    Reduce  ^x  —  4  +  T  =  10. 

operation. 


^a:  — 4  +  7  =  10 
Transposing,  a^  x  —  4  =    3 

Cubing,  a:  —  4  =  27 

Transposing,  ar  =  31 


RADICAL  EQUATIONS.  171 


3.   Keduce  ^^±^  =  sTa. 


OPERATION. 


Clearing  of  fractions,  ^  d"^  -\-  ^/  x  ^=  a 
Squaring",  c?^  +  V^  =  ^^ 

Transposing,  ^  x  =:  a^  —  d'^ 

Squaring,  xz=z{(j?  —  ^2^2 

Hence,   to   reduce    radical   equations,   we    deduce   from 
these  examples  the  following  general 

RULE. 
Transpose  the  terms  so  that  a  radical  part  shall  stand  by 
itself;  then  involve  each  member  of  the  equation  to  a  power 
of  the  same  name  as  the  root ;  if  the  unknown  quantity  is 
still  under  the  radical  sign,  transpose  and  involve  as  before ; 
finally  reduce  as  usual. 

m 

4.  Reduce  4  +  1  +  3  f^lc  =  ^{-.  Ans.  x  =  16. 

5.  Reduce  -4/-  =  -.  Ans.  x  =  2. 

4  y   X        2 

6.  Reduce   (a/Ic  +  4)^"  —  2.  Ans.  x  =  144. 


7.    Reduce  V 11  +  a;  =  \/a?  +  1.  Ans.  a?  =  25. 


8    Reduce  V^  — 7  — /v/x  + 18  — >v/5- 

Ans.  x  =  27. 

9.   Reduce  ■   ^^     =  ~ .  Ans.  x  = 

X  —  ex         ^  X  1  — <^ 

10.    Reduce  X_^:=ll  =  >^x—X  ^^^    x^^, 

V^x+lO         v^a:+23 


172  ELEMENTARY   ALGEBRA. 


11.  Reduce  s/  x  4-  \/a:  —  a  =    . 

yx  —  a 

12.  Reduce  s/ x  —  30  +  \/a:  +  21  =  ^/ x  —  19. 


13.  Reduce  /s/9a:4- 13  =  3  Va^+ 1.      Ans.  x  =  4. 

14.  Reduce   V ' —  =  V ' Ans.  a:  =  5. 

V/5a;  -|-  1         ^5x-{-  2 


15.   Reduce  V^'  — 32  =  x  —  ^V  32. 

168.    Equations    containing    the    unknown    quantity    in- 
volved to  any  power  require  Evolution  in  their  reduction. 

160.    To  reduce  pure  equations  containing  the  unknown 
quantity  involved  to  any  power. 


1.    Reduce  -^ 

3  _ 

■  7  ~~ 

97 
35 

OPERATION. 

At^           3_ 

97 

Clearing  the  given  equation  of 

5             7  ~ 

35 

fractions,  transposing,  and  divid- 

282^2—  15  = 

97 

ing,    we    have  x'  =  4;   extract- 

28x2  = 

112 

•  ing  the  square  root  of  each  mem- 

x^ = 

4 

ber   of   this   equation,    we    have 

X=z 

±2 

ar=±2.     (Art.  136.) 

2.    Reduce  Tx*  — 

-89z 

=  100. 

OPERATION. 

f  a:«  —  89  = 
7x«  = 

100 

189 

Transposing   and   dividing,  we 
have    a:*  =s   27;     extracting    the 
cube    root   of  each    member    of 

x»  = 

27 

this   equation,    we   have    x  =  3. 

X    = 

3 

Hence, 

RULE. 
Reduce  the  equation  so  as  to  have  as  one  member  ike  un- 
known quantity  involved  to  any  degree,  and  then  extract  that 
root  of  each  member  which  is  of  the  same  name  as  the 
power  of  the  unknown  quantity. 


PURE   EQUATIONS   ABOVE   THE   FIRST   DEGREE.  173 

Note.  —  It  appears  from  the  solution  of  Example  1  that  every 
pure  quadratic  equation  has  two  roots  numerically  the  same,  but  with  op- 
posite  signs. 

3.  Keduce  ^  a;^  +  T  =  ?  a;^  4-  3.  Ans.  x^  ±6. 

4.  Reduce  a =5 ^  • 

c  a 


^      Ihcd  — 


.                     .       ,  ., ^  ^       acd 
Ans.  x=z  ±  i/ ^— . 


4  j;2  24 

5.   Reduce  r =  10.  Ans.  x  =  ±  4:. 


14  1 

6.  Reduce  3a:^+ 3  =  9*  Ans,  x=-. 

7.  Reduce  --+50  =  1.  Ans.  x  =  —  1. 

8.  Reduce^! — r— =  — ^P—-       Ans.  a:=±\/ — 5. 

2ar-|-l  X  -\-4  ^ 

9.  Reduce  4a:'  — 4ar«  =  0. 

10.  Reduce  5a:2— 3a:  =  8x2  — 3a;  +  50. 

1  7 

11.  Reduce  x  -{-  -  =  -^ 1. 

12.  Reduce  2a;  +  2  =  (a:+ 1)2. 

13.  Reduce  1  +  14  a?"^  =  2  —  2  a--^. 

14.  Reduce  3  a:-^  —  5  x'^  =  2  a;-^  _|_  3  x-^  —  |. 

15.  Reduce  (c  +  a:)«  —  6  c^ar  =  (c  —  x)^  +  16  cl 

16.  Reduce^2-3^-p3  =  3^. 


n.   Reduce  ^_r  f-  +  -J^tt"  =2. 


174  ELEMENTARY  ALGEBRA. 

170.  Equations  containing  radical  quar^ities  may  re- 
quire in  their  reduction  both  Involution  and  Evolution  ; 
and  in  this  case  the  rule  in  Art.  167,  as  well  as  that  in 
Art.  169,  nnust  be  applied.  Which  rule  is  first  to  be  ap- 
plied depends  upon  whether  the  expression  containing  the 
unknown  quantity  is  evolved  or  involved. 

1.   Reduce  17  —  V^""-^^  =  12. 

OPERATIOX. 

17  —  \/;r3-^2  =  12 

Transposing,  &c.,  \/a:^  —  2  =    5 

Squaring,  a:»  —  2  =  25 

Transposing  and  uniting,  a:^  =  27 

Extracting  the  cube  root,  x  =    S 


2.   Reduce  (Va:«  —  4  +  3)«  =  125.  Ans.  x  =  2. 


3.   Reduce  i/^— ^^ —  =  Vx.  Ans.  x=  ±  - 


4.   Reduce  \/x  -{-  a  = 


\J  X  —  a 


Ans.  x—±  /v/2a-2-f2a6  +  6». 


5.  Reduce  ^-^3  (.^  +  11).^:^. 

6.  Reduce  ^'2  x*-\-%x^  +  24^^+"32~i^  =  x  +  2. 

7.  Reduce  ^9  (x*  -f  19)  +  100  —  ^  ==  ^^^ 

171.  Equations  which  contain  two  or  more  unknown 
quantities  may  require  for  their  reduction  involution,  oi 
evolution,  or  both.  In  these  equations  the  elimination  is 
effected  by  the  same  principles  as  in  simple  equations. 
(Arts.  112-114.) 


(2x2  4-v  =  64) 


1.   Given  •<(    5         4  ^,  to  find  x  and  y. 


PUKE   EQUATIONS   ABOVE   THE   FIRST   DEGREE.  175 

OPERATION. 


'f  l^u 

(1) 

2x'  +  y=    54 

(2) 

^-.^    56 

(3) 

-/  =  no 

(t) 

15-?=  14 

0) 

x2=    25 

(5) 

!/=    4 

(8) 

a;  =±5 

(6) 

Adding  four  times  (1)  to  (2),  we  obtain  (4),  which  reduced  gives 
(6),  or  X  =  -j-  5;  substituting  this  value  of  x  in  (1),  we  obtain  (7), 
which  reduced  gives  (8),  or  y  =  4. 

Find  the  value  of  the  unknown  quantities  in  the  follow- 
ing equations:  — 


2.   Given 


=  ±5. 


3.  Given    |3.-4y=2y)  ^^^     ^xr=±^. 

(^x^z=z2Q  \  {x=i  ±\. 

4.  Given    -j  2x  2r  =  10  >- -  Ans.  ^  2^  =  ±  3. 

(32^2:  =  45)  (z=±b. 

5.  Given    |i:1-^I=n.  Ac«.    j^^^^T. 

6.  Given    1-^  +  2/^  =  97  >. 

(a:   —  y  =:y  —  2x} 


7.    Given 


(x=*  — 2y2=14    ). 


176  ELEMENTABY  ALGEBRA. 


PKOBLEMS 

PRODUCING  PURE  EQUATIONS  ABOVE  THE  FIRST 
DEGREE. 

172.  Though  the  numerical  negative  values  obtained  in 
solving  the  following  Problems  satisfy  the  equations 
ibrmed  in  accordance  with  the  given  conditions,  they  are 
practically  inadmissible,  and  are  therefore  not  given  in 
the  answers. 

1.  A  gentleman  being  asked  how  many  dollars  he  had 
in  his  purse,  replied,  "  If  you  add  21  to  the  number  and 
subtract  4  from  the  square  root  of  the  sum,  the  remainder 
will  be  6/'     How  many  had  he? 

SOLUTION. 

Let  X  =  number  of  dollars. 


Then,  V«  +  21— 4=      6 

Transposing,  v'^  +  2i=    10 

Squaring,  a:  +  21  =  100 

Transposing,  x  =    T9,  number  of  dollai^. 

2.  Divide  20  into  two  parts  whose  cubes  shall  be  in 
the  proportion  of  27  to  8.  Ans.   12  and  8. 

3.  "What  two  numbers  are  those  whose  sum  is  to  the 
less  as  8  :  3,  and  the  sum  of  whose  squares  is  136  ? 

Ans.  10  and  6. 

4.  What  number  is  that  whose  half  multiplied  by  its 
third  gives  54  ? 

5.  What  number  is  that  whose  fourth  and  seventh 
multiplied  together  gives  46f  ?  Ans.  36. 

6.  There  is  a  rectangular  field  containing  4  acres  whose 
length  is  to  its  breadth  as  8:6.  What  is  its  length  and 
breadth  ? 


PURE   EQUATIONS   ABOVE   THE   FIRST   DEGREE.  177 

*J.  There  are  two  numbers  whose  sum  is  It,  and  the 
less  divided  by  the  greater  is  to  the  greater  divided  by 
the  less  as  64  :  81,     What  are  the  numbers? 

Ans.   8  and  9. 

8.  The  sum  of  the  squares  of  two  numbers  is  65,  and 
the  difference  of  their  squares  33.     What  are  the  numbers  ? 

9.  The  sum  of  the  squares  of  two  quantities  is  a,  and 
the  difference  of  their  squares  b.     What  are  the  quantities  ? 

Ans.    ±  \/i  {a  +  b)  and   ±  \/ ^  {a  —  b). 

10.  A  gentleman  sold  two  fields  which  together  con- 
tained 240  acres.  For  each  he  received  as  many  dollars 
an  acre  as  there  were  acres  in  the  field,  and  what  he 
received  for  the  larger  was  to  what  he  received  for  the 
smaller  as  49:25.     What  are  the  contents  of  each? 

Ans.   Larger,  140;  smaller,  100  acres. 

11.  What  are  the  two  quantities  whose  product  is  a 
and  quotient  6?  —  /"^ 

Ans.   ±  V  «  6  and   ±  i  /  t  • 

12.  What  two  numbers  are  as  m  :  n,  the  sum  of  whose 

squares  is  a  ?  msia  ,     ,  ni/a 

Ans.   ±  -. —  v__  and  ±  -, — -^ 

13.  What  two  numbers  are  as  m:n,  the  difference  of 
whose  squares  is  a  ?  ,—  ,-- 

Ans.   db  ~, and  ±  ^ 


14.  Several  gentlemen  made  an  excursion,  each  taking 
$484.  Each  had  as  many  servants  as  there  were  gentle- 
men, and  the  number  of  dollars  which  each  had  was  four 
times  the  number  of  all  the  servants.  How  many  gen- 
tlemen were  there?  Ans,   11. 

15.  Find  three  numbers  such  that  the  product  of  the 
first  and  second  is  12  ;  of  the  second  and  third,  20  ;  and 
the  sum  of  the  squares  of  the  first  and  third,  34. 

8*  L 


178  ELEMENTARY   ALGEBRA. 

SECTION   XIX. 

AFFECTED    QUADRATIC    EQUATIONS. 

173.  An  Affected  Quadratic  Equation  is  one  that  con- 
tains both  the  first  and  second  powers  of  the  unknown 
quantity  ;  as, 

3  x^  —  4  ar  =:  16  ;  or  a  a;  —  bx^  ■=€. 

174.  Every  affected  quadratic  equation  can  be  reduced 
to  the  form 

x^  -\-bx=zc, 

in  which  b  and  c  represent  any  quantities  whatever,  posi- 
tive or  negative,  integral  or  fractional. 

For  all  the  terms  containing  oc^  can  be  collected  into  one  term 
■whose  coefficient  we  will  represent  by  a ;  all  the  terms  containing  x 
can  be  collected  into  one  terra  whose  coefficient  we  will  represent 
by  d\  and  all  the  other  terms  can  be  united,  whose  aggregate  we 
will  represent  by  e.  Therefore  every  affected  quadratic  equation 
can  be  reduced  to  the  form 

a3^-\-(lx  =  e         (1) 

Dividing  (1)  by  a,  a:^  -f-  ^x  =  ^         (2) 

d  e 

Letting  -  =  h,  and  -  =  c,  we  have         x^  -\-  bx  ^  c        (3) 

175.  The  first  member  of  the  ecjuation  x^  -{-  bx  z=  c 
cannot  be  a  perfect  square.  (Art.  145^  Note  2.)  But 
we  know  that  the  square  of  a  binomial  is  the  square  of 
the  first  term  plus  or  minus  twice  the  product  of  the  tuv 
terms  plus  the  square  of  the  last  term;  and  if  we  can 
find  the  third   term  which  will   make  x^  -\-  bx   a  perfect 


EQUATIONS   OF   THE   SECOND   DEGREE.  179 

square  of  a  binomial,   we  can  then    reduce  the    equation 

Since  h  x  has  in  it  as  a  factor  the  square  root  of  a:^,  x^  can  be 
the  first  term  of  the  square  of  a  binomial,  and  h  x  the  second  term 
of  the  same  square;  and  since  the  second  term  of  the  square  is 
twice  the  product  of  the  two  terms  of  the  binomial,  the  last  term  of 
the  binomial  must  be  the  quotient  arising  from  dividing  the  second 
term  of  the  square  by  twice  the  square  root  of  the  first  term  of  the 

*  square  of  the  bino- 

mial; i.  e.  the  last 

OPERATION.  .  /•  ^1,      u- 

term  of  the  bmo- 

x^-\-hx  =  c  (1)  ,  ^    .     hx        h 

vaidl  is    --  =  -  ; 

and  therefore  the 
third  term  of  the 


+  *^  +  4-=4+<'  (2) 


^  +  l  =  ±v/r  +  «  (3) 


i  /  T"  ~r  ^  \^)  square    must    be 

(2)^4- ^"^'^^"S 


=  -2^   V4+^ 


(4) 


—  to  each 
4 


ber,  we  have  (2),  an  equation  whose  first  member  is  a  perfect 
square.     Extracting   the   square  root   of  each  member  of  (2),  and 

transposing,  we  obtain  (4),  or  x  =  —  9 -"-  v/ aT  "^  ^'  ^^'^^^  ^®  ^ 
general  expression  for  the  value  of  x  in  any  equation  in  the  form 
of  a:^  -f"  ^  ^  =  *^' 

Hence,  as  every  affected  quadratic  equation  can  be  re- 
duced to  the  form  x"^  -]-  bx  =  c,  in  which  b  and  c  repre- 
sent any  quantities  whatever,  positive  or  negative,  integral 
or  fractional,  every  affected  quadratic  equation  can  be  re- 
duced by  the  following 

RULE. 

Reduce  the  equation  to  the  form  x^  -\-  bx  =  c,  and  add 
to  each  member  the  square  of  half  the  coefficient  of  x. 

Extract  the  square  root  of  each  member,  and  then  reduce 
as  in  simple  equations. 


180  ELEMENTARY   ALGEBRA. 

1.    Reduce  Tx2  — 28  ar+ 14  =  238. 

OPERATION. 

Tar*— 28 ar  4- 14  =  238 
Transposing",  7  a:^  —  28  x  =  224 

Dividing  by  7,  x^  —    4x=    32 

Completing  the  square,      x^  —  4ar  -|-  4  =    36 
Evolving,  X  —  2  =±6 

Transposing,  a:=:2±6  =  8,  or  —  4 

Note.  —  Since  in  reducing  the  general  equation  a:^  -|-  6  x  =  c 
we  find  x  =  —  oil/j  ~l~<^»  every  affected  quadratic  equation 
must  have  two  roots;  one  obtained  by  considering  the  expression 
-  -|-  c  positive,  the  other  by  considering  this  expression  nega- 


tive.    Whenever  4  /  -  -[-  c  =  0  these  two  roots  will  be  equal. 


2.    Reduce  -— --U-^'^-l-f. 

5  10    '     20         2     "^  4 


OPERATION. 

^  ^       1^    13  Tr     I     X 

5  10'"   2"0         2"'~4 


Clearing  of  fractious,  4x'^  —  2  a?  -|-  13  =  10 a:^  -{-  bx 
Transposing,  — 6a:'^  —  7ar  =  —  13 


Dividing  by  —  6, 

x'  + 

Ix  _ 
6 

_  13 

Completing  the  square, 

^'  +  ()  + 

40  _ 
144" 

_   49 
"  144 

:  + 

is  _ 

6   ~ 

Evolving, 

■^  + 

7 
12 

=  ± 

19 
12 

Transposing,                 x 

—k^ 

19   _ 
ll  ~ 

=  1, 

or- 

-H 

861 
144 


EQUATIONS  OF  THE  SECOND  DEGREE.        181 

]*foTE. — In  completing  the  square,  as  the  second  term  disappears 
when  the  root  is  extracted,  we  have  written  (  )  in  place,  of  it. 

3.  Reduce  Sx^  —  25 -\- 6x  =  SO. 

Ans.  X  =^  5,  or  —  T. 

4.  Reduce  x =  3.       Ans.  x  =  6,  or  —  4. 

X 

5.  Reduce  2x  +  — ^  =  1.  Ans.  x  =  2, 

'    X  —  1 

Note.  —  In  this  example  both  roots  are  2. 


x"  -I-  4 

6.   Reduce  1  x -^  =  5  x  —  1. 

X  —  4 


Ans.  X  =  8,  or  —  1. 


H      r>    J  iH         U—x         IS  —  X    . 

1.   Reduce  n ^ — —  g  _  ^  +  ^^- 

Ans.  a:  =  t,  or  —  21. 

8.  Reduce      ^      -{-  -=  ^.     Ans.  a;  =  10,  or — 1§. 

X  — j—  O  o  10 

9.  deduce  '-'  +  '-'^^  =  i. 


,^      _,    -  16         100  — 9a: 

10.    Reduce —^ —  =  3. 

X  4ar  ' 


176.  Whenever  an  equation  has  been  reduced  to  the 
form  x"^  -{-  bx  =  c,   its  roots  can  be  written   at  once;  for 

this  equation  reduced  (Art.  175)  gives  x  = —  -  ±  » /-  -\-c. 
Hence, 

ITw  roots  of  an  equation  reduced  to  the  form  x'^  -\-  bx  =  c 
are  equal  to  one  half  the  coefficient  of  x  with  the  opposite 
sign,  plus  or  minus  the  square  root  of  the  sum  of  the  square 
of  one  half  this  coefficient  and  the  second  member  of  the 
equation. 


182  ELEMENTARY  ALGEBRA. 

In  accordance  with  this,  find  the   roots  of  x  in  the  fol- 
lowing equations :  — 

1.    Reduce  x^^  8x==65. 


a:  =  —  4  ±  V 16  +  65  =  5,  or  —  13,  Ans. 
2.   Reduce  a:^  —  lOa;  =  —  24. 


ar  z=  5  ±  yv/25  —  24  =  6,  or  4,  Ans. 

3.  Reduce  a;^  _  6  x  ==  —  5.  Ans.  or  =  5,  or  1. 

4.  Reduce  a:^  -|-  7  a:  =  170. 

7  /  49 

5.  Reduce  a;'*  -f-  9^  =  9- 


=-5  ±\/r6  +  ^  =  i  °"  "-^' 


Ans. 


1  9 

6.    Reduce  x'^  +  -a:  =-•  Ans.  a:  =1,  or — IX. 

00 


•7        -D     J                   2               ^                             ^  A                              3                  1 

7.  Keauce  x^  —  7^=  — ;;^-  Ans.  a:  = ->    or  -• 

5                    2d  5            5 

8.  Reduce  -  =  -  -f-  6^.  Ans.  a:  =  7,  or  —  5^. 


SECOND  METHOD   OF  COMPLETING  THE   SQUARE. 

177i  The  method  already  given  for  completing  the 
square  can  be  used  in  all  cases  ;  but  it  often  leads  to  in- 
convenient fractions.  The  more  difficult  fractions  are  in- 
troduced by  dividing  the  equation  by  the  coefficient  of 
a:^  to  reduce  it  to  the  form  x'^-\-hx=zc.  To  present  a 
method  of  completing  the  square  without  introducing  these 
fractions,  we  will  reduce  equation  (1)  in  Art.  174. 


EQUATIONS   OF   THE   SECOND  DEGREE.                      183 
1.    Reduce  ax^  -\-  dx  •=  e. 

OPERATION. 

ax^  -\-  dx  =  e  (1) 

a^x^-\-adx=zae  (2) 

a'x'^  +  adx  +  '^  =  '^-^  +  ae  (3) 


ax  At  2  =  ±  y  7  +  ««  W 

Multiplying  (1)  by  a,  the  coefficient  of  x^,  we  obtain  (2),  in  which 

the  first  term  must  be  a  perfect  square.     Since  ad x^  the   second 

term,  has  in  it  as  a  factor  the  square  root  of  a^  x^^  a?  x^  can  be  the 

first  term  of  the  square   of  a  binomial,  and  ad x  the  second  term ; 

and  since  the  second  term  of  the  square  is  twice  the  product  of  the 

two  terms  of  the  binomial,  the  last  term  of  the  binomial  must  be  the 

second  term  of  the  square  divided  by  twice  the  square  root  of  the  first 

d  (1  jc       d 
term  of  the  square  of  the  binomial,  or  ^ —  =  - ;  and  therefore  the 

term  required  to  complete  the  square  is  — ,  which  is  the  square  of 

one  half  of  the  coefficient  of  x  in  (1).  Adding  —  to  both  members 
of  (2),  we  obtain  (3),  whose  first  member  is  the  square  of  a  binomial. 
Extracting  the  square  root  of  (3)   and  reducing,  we  obtain   (5),  or 


1/       '^o.      /^'j        \ 


Hence,  to  reduce  an  affected  quadratic  equation,  we 
have  this  second 

RULE. 

Reduce  the  equation  to  the  form  ax^  ~\-  dx  =  e ;  then  mul- 
tiply the  equation  by  the  coefficient  of  x^,  and  add  to  each 
member  the  square  of  half  the  coefficient  of  x. 

Extract  the  square  root  of  each  member,  and  then  reduce 
as  in  simple  equations. 


184  ELEMENTARY  ALGEBRA. 

Note  1.  —  This  method  does  not  introduce  fractions  into  the  equa- 
tion when  the  numerical  part  of  the  coefficient  of  x  is  even.  When 
the  coefficient  of  7?  is  unity,  this  method  becomes  the  same  as  the 
first  method. 

Note  2.  —  If  the  coefficient  of  j?  is  already  a  perfect  square  the 
square  can  be  completed  without  multiplying  the  equation,  by  add- 
ing to  both  members  the  square  of  the  quotient  arising  from  dividing 
the  second  term  by  twice  the  square  root  of  the  first.  This  method 
also  becomes  the  same  as  the  first  method  when  the  coefficient  of 
a^  is  unity. 

Note  3.  —  As  an  even  root  of  a  negative  quantity  is  impossible 
or  imaginary,  the  sign  of  the  first  term,  if  it  is  not  positive,  must 
be  made  so  by  changing  the  signs  of  all  the  terms  of  the  equation. 

2.  Reduce  3  x^  +  8  a:  =  28. 

OPERATIOIf. 

3a:2+8a;  =  28 
Completing  the  square,       9  x^  +  ( )  +  16  =  16  +  84  =  100 
Extracting  square  root,  Sx-\-4:=  ±  10 

Whence,  3x=:  — 4  ±10  =  6,  or— 14 

And  x  =  2,  or  —  4§ 

3.  Reduce  252?^  —  10  a:  =  195. 

OPERATION. 
25x^—  10x=:195 
Completing  sq.  by  Note  2,   25x^—  ( )  +  1  =  1  +  195  =  196 
Extracting  square  root,  bx —  1  =  ±  14 

Whence,  6a:=l  ±  14=15,  or—  13 

And  x=   3,  or  — 2a 

4.  Reduce  5  a:'^  —  20  a:  =  —15.  Ans.  a?  =  3,  or  1. 
6.    Reduce  7a:2  — 8a:=  12f.           An8.  x  =  ^  ±  ^V^'^- 

6.  Reduce  7  x^  —  4  a  a:  =  -,  •        Ans.  x  =  ~   ,  or  —  -  • 

/  7  7 

7.  Reducer- ^^I— =a; — 3.    Ans.  a;=  10,  or  3J. 

14  —  X  o 


EQUATIONS  OF  THE  SECOND  DEGREE.        185 
8.  Reduce  x^  -j —  = 4. 


4  4 

r'  — 4         X— 3         3a:— 7 


9.   Reduce 


THIRD  METHOD  OF  COMPLETING  THE   SQUARE. 

178.  The  method  of  the  preceding  Article  iutroduces 
fractions  whenever  the  numerical  coefficient  of  x  is  not 
even.  To  present  a  method  of  completing  the  square 
without  introducing  any  fraction,  we  will  again  reduce 
equation  (1)  in  Art.  IH. 


1.    Reduce  a:/?  -\-  dx^^e. 


OPERATION. 

ax^  -\-  dx=:.e 

(1) 

„    ,     d             e 
x'^-\-  -  X  =- 

(2) 

^  _.     d      j_    (P           ^_L^ 
^  ■T"a^'+"4a^  —  4a^"^"a 

(3) 

4:a^a^  +  4:adx  +  cP  =  d^-\-4.ae 

(4) 

2ax  +  d=  ±  ^d'--^^ae 

(5) 

—  d±  v/tP-f-4ae 
^  — 

(6) 

Dividing  (1)  by  a,  the  coefficient  of  oc^,  we  have  (2)  ;  then  com- 
pleting the  square  according  to  the  Rule  in  Art.  1 75,  we  have  (3) ; 
and  if  we  multiply  (3)  by  4  a^  it  will  give  (4),  an  equation  free 
from  fractions  (unless  a,  d,  or  e  in  (1)  are  themselves  fractions), 
and  one  whose  first  member  is  the  square  of  a  binomial.  To  pro- 
duce this  equation  directly  from  (1),  we  have  only  to  multiply  (1) 
by  4  a ;  i.  e.  by  four  times  the  coefficient  of  3^,  and  add  to  both 
members  d^;  i.  e.  the  square  of  the  coefficient  of  x.  Reducing  we 
have  (6),  which  is  a  general  expression  for  the  value  of  x  in  any 
equation  in  the  form  of  a  r*  -j-  rf  ar  =  c. 


186  KLEMENTARY   ALGEBRA. 

Hence,  to  reduce  an  affected  quadratic  equation,  we 
have  this  third 

RULE. 

Reduce  the  equation  to  the  form  ax^  -\-  dx  :^  e ;  then  mul- 
tiply the  equation  by  four  times  the  coefficient  of  x^  and  add 
to  each  member  the  square  of  the  coefficient  of  x. 

Extract  the  square  root  of  each  member,  and  then  reduce  as 
in  simple  equations. 

Note.  —  The  third  Note  under  the  Rule  in  Art.  177  is  applica- 
ble in  all  cases. 


2. 

Reduce  b  x^  - 

-1x  =  24:. 

OPERATION. 
5a:2  — 7ar  =  24 

Multiplying  by  5  X  4  and 

adding  7^  to  each  member,  lOOx^  —  (  )  +  49  =  49  +  480  =  529 
Extracting  the  square  root,  lOx  —  7  =  ±23 

Transposing,  10  a:  =  7  ±  23  =  30,  or  —  16 

Whence,  a:  =  3,  or  —  1.6 

Note.  —  The  multiplication  of  the  coefficient  of  s^  need  only  be 
expressed.  Its  coefficient  after  evolving  is  double  its  original  coef- 
ficient. 

3.    Reduce  ^^^— ^^a:  _  ^^^ 

OPERATION. 
44x2— 15a: 


=  293 


7 

Clearing  of  fractions,  44  x^  —  1 5  x  =  2051 

Completing  square,  176  X  44x2  —  {)  -f  225  =  225  +  360976  =  361201 
Evolving,  88  X  —  15  =  ±  601 

Transposing,  88  x  =  15  ±  601  =  616,  or —  586 

Whence,  x  =  7,  or 

44 

Jr.    Reduce  Y  a;*  —  15  x  =  —  2.  Ans.  a:  =  2,  or  ^. 


EQUATIONS   OF  THE   SECOND  DEGREE.  187 


6. 

Reduce 

Ans.  X  =z 

1,  or  — 

■28, 

6. 

Reduce 

10        1     9  _ 

5.           Ans.  a:  = 

3,  or  — 

■H 

1, 

Reduce 

a:+  1    '    a; 

:  3.            Ans.  x  = 

=  2,  or  - 

-i 

8. 

Reduce 

4  a: -f- .4         Sx 

—  3  __  lOar-f  10 

X               2x 

—  1  ~         3x 

9. 

Reduce 

1    + 

7  —  2a;    "^  2x 

3        _  13 

4-4  ~  10* 

10. 

Reduce 

^^3-_ra^  = 

■.x  —  b. 

Ans.  X  =  -  ± 

V       126    ' 

11. 

Reduce  b  —  3x-^=z 

UOx-\ 

Note.  —  Mult 

iply  by  3^. 

12. 

Reduce 

^-^  _|_  ^^3 

=  6  V^. 

Note.  —  Divide  by  \/  x. 

179.  The  rules  which  have  been  given  for  the  solution 
of  affected  quadratic  equations  apply  equally  well  to  any 
equation  containing  but  two  powers  of  the  unknown  quan- 
tity whenever  the  index  of  one  power  is  exactly  twice  that 
of  the  other.  By  the  same  reasoning  as  in  Art.  174,  it  can 
be  shown  that  all  such  equations  can  be  reduced  to  the 
form 

ax'^''  -^  dx""  =  e, 
or 

It  will  be  seen  that  the  first  member  is  composed  of 
two  terms  so  related  that  they  may  be  the  first  two  terms 
of  a  binomial  square,  and  we  can  supply  the  third  by  one 
of  the  rules  already  given  for  completing  the  square. 


188  ELEMENTARY  ALGEBRA. 

1.  Reduce  a:«  —  2^^  =  48. 

OPERATION.  O- 

oince  the  square  root 

x'  —  2x'  =  4.S  (1)  of  a:«  is  a:»,  it  is  evident 

a;®  —  2a?^  +  1  =  1  +  48  =:  49     (2)  that  the  second  terra 

_8        1  I    >T  /o\  contains  as  one  of  its 

^  —  1  =  ±  7  (3)  -    ^ 

lactors  the  square  root 

a^  z=  8,  or  —  Q__        (4)  of  the  first  term ;  i.  e. 

X  =z  2,  or  ^ —  6       (5)  the  first  member  of  the 

equation  is  composed 
of  two  terms  so  related  that  they  may  be  the  first  two  terms  of  the 
square  of  a  binomial.  Completing  the  square,  we  have  (2)  ;  extract- 
ing the  square  root  of  each  member  of  (2),  we  obtain  (3)  ;  transpos- 
ing we  have  (4),  and  extracting  the  cube  root  of  (4)  we  have  a;  =  2, 
or  ^^^ 

2.  Reduce  3a;^  — 4a:^=  160. 

OPERATION. 

3a;*  — 4a;^=160  (1) 

36  x*  ~  (  )  +  16  =  16  +  1920  =  1986  (2) 

6  a;^  —  4  =  ±  44  (3) 

Q'x^  =  48,  or  —  40  (4) 

x^  =    8,  or  —  5j0-  (5) 

J 


=    2,  or^-^-  (6) 

a:=  16,  or  (— ^3<i)^  (7) 

In  this  equation  the  index  of  the  higher  power  is  exactly  twice 
that  of  the  lower.  Completing  the  square  we  have  (2)  ;  extract- 
ing the  square  root  of  each  member  of  (2),  we  have  (3)  ;  trans- 
posing, we  have  (4),  which  divided  by  6  gives  (5) ;  extracting  the 
cube  root  of  (5),  we  have  (G),  which  involved  to  the  fourth  power 
gives  (7). 


af"    ,    ac"         3 
32 


3.    Reduce  --!--=— .      Ans.  a:  = -^^,  or  —  >C^^. 

Z  4  32 


4.   Reduce  s/ar*  4-^-^ a:  =:  1.     Ans.  a:  =  |,  or  —  8. 


EQUATIONS   OF  THE   SECOND  DEGREE.  189 

5.  Reduce  x  —  §  \/x  =  44^.      Ans.  x  =  49,  or  40^. 

6.  Reduce  x^  —  x^  =  0.  Ans.  x  =  1,  or  0. 
Y.    Reduce  3x^  —  2x^  +  3  =  228. 


Ans.  X  =  ±  S,  or  ±  5  \/  —  ^. 
8.    Reduce  3  ^r^"  —  2  ^r'*  =  8. 


Ans.  a;  =  /</  2,  or  -</  —  |. 

9.    Reduce  - — -^^'— -  =      '  -1--  •       Ans.  x  =  64,  or  4. 
4  -j-  V^^  y'a: 

10.    Reduce  x^  —  ax^  z=b. 


Ans.x  =  ±y/(|±^^'  +  *). 


180.  A  polynomial  may  take  the  place  of  the  unknown 
quantity  in  an  affected  quadratic  equation.  In  this  case 
the  equation  can  be  reduced  by  considering  the  polyno- 
mial as  a  single  quantity. 

1.    Reduce  {x  —  4)^  —  2  (ar  —  4)  =  8. 

OPERATION. 

(a:_4)2  — 2(x  — 4)  =  8       .  (1) 

(;,_4)^_()  +  l^l_|.8  =  9  (2) 

X  — 4  —  1=  ±  3  (3) 

a;  =  5  ±  3  =  8,  or  2  (4) 

Considering  x  —  4  as  a  single  term,  and  completing  the  square, 
we  have  (2) ;  extracting  the  square  root,  transposing,  &c.,  we  have 
(4),  or  a:  =  8,  or  2. 

Note.  —  We  might  put  {x  —  A)  =  y,  then  {x  —  4)"  =  y^,  and 
the  equation  becomes  y^  —  2  3/  =  8.  After  finding  the  value  of  y 
in  this  equation,  x  —  4  must  be  substituted  for  y. 


2.    Reduce  \/5  +  a:  +  ></5  +  xr=6. 

Ans.  x  =  11,  or  T6. 


3.    Reduce  i-{-2x  — x^  +  ^^4r-\-2x  — x^  =  ^. 

Ans.  a:  =  1  ±  ^ \/^,  or  3,  or  —  1. 


190  ELEMENTARY  ALGEBRA. 


4.   Keduce  x  -\- 1  —  1  f^ x  +  7  =  8  —  5\/^  +  T. 

Ans.  a:  =  9,  or  —  3. 

40 


5.    Reduce  (x  —  6)"  —  Zs/ x  —  6 

^  '  ^  X  —  5 

Ans.  ar  =  9,  or  5  +  v^25. 


6.   Reduce  x^-\-Zx-\-  s/ x'  -|-  3a:  +  6  =  14. 
Note.  —  Add  6  to  both  members. 

Ans.  a:  =  2,  or  —  5,  or  —  f  ±\  V  85. 


%.    Reduce  4  +  a:2  — 2a:  — 2  V6  — 2a:  +  a:=^  =  J^^_ 
Ans.  a:  =  3,  or  —  1,  or  1  ±  2  \/ —  1. 


60 


8.    Reduce  /\/  x'^  +  a:  +  6  =    ...  _, __ 4. 

Ans.  a:  =  5,  or  —  6,  or  —  ^  ±  J  \/ 377. 

181 1  Of  the  methods  given  for  completing  the  square, 
the  first  is  the  best  when  the  coefficient  of  the  less  power 
of  the  unknown  quantity  is  even,  and  the  coefficient  of  the 
higher  power  is  unity,  or  when  these  become  so  by  reduc- 
tion ;  the  second  method  is  better  than  the  third  when- 
ever the  coefficient  of  the  less  power  of  the  unknown  quan- 
tity is  even.  When  the  equation  cannot  be  reduced  by 
the  first  method  without  introducing  fractions,  if  the  co- 
efficient of  the  higher  power  of  the  unknown  quantity 
is  a  perfect  square,  and  the  coefficient  of  the  less  power 
is  divisible  without  remainder  by  twice  the  square  root 
of  the  coefficient  of  the  higher  power,  the  method  given 
in  Note  2,  Art.  177,  is  the  best.  Let  each  of  the  follow- 
ing equations  be  reduced  by  the  method  best  adapted 
to  it. 

1.    Reduce  4a:^— 14  =  3a:2— 12a;— 1. 

Ans.  a:  =  1,  or  —  13. 

-      2.    Reduce  36  a:* +  24  ar=  1020. 

Ans.  a:  =  5,  or  —  5§. 


EQUATIONS   OF  THE   SECOND  DEGREE.  191 


3. 

Reduce  x  —  - 

+ 

^-  —  "IX.           Ans.  a:  =  21,  or  0^. 

X      '                                                           '           . 

4. 

x—2         2a:— 11 

6       ~     x  —  3 

5. 

Reduce  -  —  - 
2          5 

— 

fo  =  -- 

6. 

_,    J           X  —  3c 
Reduce  — ^ — 

= 

9(d-c) 

X 

Ans.  a:  =  3  (c  —  rf),  or  3  d. 

1.   Reduce  ^±4  =  ^rr^  +  2|. 

o      -n    J  3a:  — 4         _  a:—  2 

8.  Reduce =  9 -— . 

X  —  4  2 

9.  Reduce 


X  9  x^ 


X    ,    a        2 


10.    Reduce  -  +  -  =  -•  Ans.  a:  =  1  ±  Vl  —  a^ 

a    ^    X        a  ^ 


11.  Reduce  3a:  +  3  i=  13  +  -• 

12.  Reduce  5  a; _      =  2  x  +  — ^ — 

io  T>    J  a:-4-4         4a:-f-7     ,     ,  7 — x 

13.  Reduce  -^t _r_  +  i  ^  ___. 


14.  Reduce  2  V^  —  V^ — 7  =  5. 

Ans,  X  =  16,  or  7^. 

15.  Reduce  2 \/T:^:^  +  3 ^/2lc  =  LliJ"^. 

\  X  —  a 

Ans.  a:  =  9  a,  or  —  a. 
15 


16.    Reduce  4\/x  — \/2a;+ 1  =  , 

'  v^2a:4-l 


Ans.  a:  =  4,  or  —  2f . 


17.    Reduce  5V25  — a;  =  6V25  —  a;  +  a:— 13. 

Ans.  X  =  16,  or  9. 


192  ELEMENTARY  ALGEBRA. 

18.  Reduce  ^-i^  =  Vi+  \/2^^^. 

Alls.  X  =  12,  or  4. 

19.  Reduce  6  +  4a:-i  —  I'ioj-^  =  lOOar-^. 

20.  Reduce  i  ^x' +  6  ^ x  — '-^  =  -• 

21.  Reduce  30:4  —  24^72  —  80  =  304. 


22.    Reduce  ^'  +  10  =  1  +  4a:». 


Ans.  a:  =  ±  4,  or  ±  2  V—  2. 

:*. 

Ans.  X  =  ^'9,  or  ^. 


23.  Reduce  5;r4  —  3a:2  4-  —  =  27. 

'      4 

Ans.  X  =  ±  J\/6,  or  ±  Sy/ —  ^, 

24.  Reduce  2x^  —  5a:^  +  4  =  2. 

25.  Reduce  6a:^+ 1184  =  5a;i 


26.    Reduce  V^  + 3 —  -^x  + 3  =  2. 

Ans.  X  z=  13,  or  —  2. 


27.    Reduce  x^  —  Vic^  +  a:  —  5  =  25  —  a:. 

Note.  —  By  transposing  —  x  and  subtracting  5  from  each  mem. 
ber,  make  the  expression  without  the  radical  in  the  first  member  like 
that  under  the  radical;  then  complete  the  square,  &c. * 


28.  Reduce  x^— 2x-{-Hx^2x^  — 6x— ll=x-\-^3. 

Ans.  a:  =  6,  or  —  3^  or  J  ±   jj^  V273. 

29.  Reduce  21  x^  _|-  U  a:^  _  69  =  321  —  (11  x*  +  6ar»). 

Ans.  x=  ±  ^\/iS,  or  ±  ^A^—i5. 

30.  Reduce  ^^^,==^  +  ^^.. 

31.  Reduce  (x2-_4a:)2z=12a:  — 3x«. 

32.  Reduce  ar  +  (ar»  — .  a:)^  =  ar»  -f  51 12. 


EQUATIONS  OF  THE  SECOND  DEGREE.        193 

PROBLEMS 

PRODUCING  AFFECTED    QUADRATIC   EQUATIONS   WITH 
BUT   ONE   UNKNOWN  QUANTITY. 

182.  Though  the  numerical  negative  values  obtained  in 
solving  the  following  Problems  satisfy  the  equations  formed 
in  accordance  with  the  given  conditions,  they  are  prac- 
tically inadmissible,  and  are  therefore  not  given  in  the 
answers. 

1.  Divide  40  into  two  parts  such  that  the  sum  of  their 
squares  shall  be  1042. 

SOLUTION. 

Let  X  =  one  part ; 

then  40  —  x  =  other  part 

Then,  a:^  -|-  (40  —  xf  =  1042 

Expanding,  a:«  -|-  1600  —  80  x  -f-  a:^  ^  1042 

Transposing  and  uniting,  a^  —  40  a;  =  —  279 

Whence,  a;  =  20  ±  11  =  31,  or  9 

And,  40  —  a;  =  9,  or  31 

2.  Divide  20  into  two  parts  such  that  their  product 
will  be  99|.  Ans.  9^  and  10^-. 

3.  The  ages  of  two  brothers  are  such  that  the.  age  of 
the  elder  plus  the  square  root  of  the  age  of  the  younger 
is  22  years,  and  the  sum  of  their  ages  is  34  years.  What 
is  the  age  of  each  ?  Ans.  Elder,  18  ;  younger,  16. 

Note.  —  The  other  answers  found  by  reducing  the  equation,  viz. 
25  and  9,  satisfy  the  conditions  of  the  equation  only  upon  consid- 
ering y/  9  =  —  3.  To  make  the  problem  correspond  to  these  an- 
swers, the  word  "plus"  must  be  changed  to  "minus." 

4.  A  merchant  had  two  pieces  of  cloth  measuring  to- 
gether 96  yards.     The  square  of  the  number  of  yards  in  the 


194  ELEMENTARY    ALGEBRA. 

longer  is  equal  to  one  hundred  times  the  number  of  yards 
in  the  shorter.     IIow  many  yards  are  there  in  each  piece  ? 

Ans.  60  and  36. 
6,    Find  two   numbers   whose   difference   is   3,    and   the 
sum  of  whose  squares  is  117.  Ans.  9  and  6. 

6.  A  merchant  having  sold  a  piece  of  cloth  that  cost 
him  $42,  found  that  if  the  price  for  which  he  sold  it  were 
multiplied  by  his  loss,  the  product  would  be  equal  to  the 
cube  of  the  loss.     What  was  his  loss  ? 

Note.  —  If  the  word  "  loss "  were  changed  to  gain,  the  other  an- 
swer, —  7,  or  as  it  would  then  become,  -\-  7,  would  be  correct. 

Ans.   $6. 

t.  Find  two  numbers  whose  difference  is  5,  and  prod- 
uct 176.  Ans.  11  and  16. 

8.  There  is  a  square  piece  of  land  whose  perimeter  in 
rods  is  96  less  than  the  number  of  square  rods  in  the 
field.     What  is  the  length  of  one  side  ?      Ans.  12  rods. 

9.  Find  two  numbers  whose  sum  is  8,  and  the  sum  of 
whose  cubes  is  152. 

10.  A  man  bought  a  number  of  sheep  for  $240,  and 
sold  them  again  for  $6.75  apiece,  gaining  by  the  bargain 
as  much  as  5  sheep  cost  him.  IIow  many  sheep  did  he 
buy?  Ans.  40. 

11.  Find  two  numbers  whose  difference  is  4,  and  the 
sum  of  whose  fourth  powers  is  1312. 

Note.  —  Let  x  —  2  and  x  -|-  2  be  the  numbers. 

Ans.  2  and  6. 

12.  A  man  sold  a  horse  for  $312.50,  and  gained  one 
tenth  as  much  per  cent  as  the  horse  cost  him.  How 
much  did  the  horse  cost  him?  Ans.  $250. 

13.  The  difference  of  two  numbers  is  5,  and  the  less 
minus  the  square  root  of  the  greater  is  7.  What  are  the 
numbers?  Ans.   II  and  16. 


EQUATIONS   OF   THE    SECOND    DEGREE.  195 

14.  A  and  B  started  together  for  a  place  300  miles  dis- 
tant. A  arrived  at  the  place  7  hours  and  30  minutes  be- 
fore B,  who  travelled  2  miles  less  per  hour  than  A.  IIow 
many  miles  did  each  travel  per  hour? 

Ans.   A,  10  ;  B,  8  miles. 

15.  A  gentleman  distributed  among  some  boys  $15; 
if  he  had  commenced  by  giving  each  10  cents  more,  5 
of  the  boys  would  have  received  nothing.  How  many 
boys  were  there  ?  Ans.  30. 

16.  Find  two  numbers  whose  sum  is  a,  and  product  b. 

a  ±  yJ  a-  —  Ah         ,   a  T  \/  d^  —  4  6 
Ans. ^~~ ■  ^^d  ^  ^ -• 

n.    A  merchant  bought  a  piece  of  cloth  for  $45,   and 

sold  it  for  15  cents  more  per  yard  than  he  paid.     Though 

he   gave    away    5    yards,    he    gained    $4.50    on  the   piece. 

How  many  yards  did  he  buy,  and  at  what  price  per  yard  ? 

Ans.  60  yards,  at  75  cents  per  yard. 

18.  A  certain  number  consists  of  two  figures  whose 
sum  is  12  ;  and  the  product  of  the  two  figures  plus  16  is 
equal  to  the  number  expressed  by  the  figures  in  inverse 
order.     What  is  the  number  ?  Ans.  84. 

19.  From  a  cask  containing  60  gallons  of  pure  wine  a 
man  drew  enough  to  fill  a  small  keg,  and  then  put  into 
the  cask  the  same  quantity  of  water.  Afterward  he  drew 
from  the  cask  enough  to  fill  the  same  keg,  and  then  there 
were  41 1  gallons  of  pure  wine  in  the  cask.  How  much 
did  the  keg  hold  ?  Ans.   10  gallons. 

20.  There  is  a  rectangular  piece  of  land  75  rods  long 
and  65  rods  wide,  and  just  within  the  boundaries  there  is 
a  ditch  of  uniform  breadth  running  entirely  round  the 
land.  The  land  within  the  ditch  contains  29  acres  and 
96  square  rods.     What  is  the  width  of  the  ditch  ? 

Ans.  .5  of  a  rod. 


196  ELEMENTARY   ALGEBRA. 


SECTION   XX. 

QUADRATIC    EQUATIONS    CONTAINING    TWO 
UNKNOWN   QUANTITIES. 

183t  The  Degree  of  any  equation  is  shown  by  the  sum 
of  the  indices  of  the  unknown  quantities  in  that  term  in 
which  this  sum  is  the  greatest.     Thus, 

4:xy  —    2x  =  7     is  an  equation  of  the  second  degree, 
hx^y'^-\-  xy^=^a^c     "  ''  "       fourth        " 

.  Note.  —  Before  deciding  what  degree  an  equation  is,  it  must  be 
cleared  of  fractions,  if  the  unknown  quantities  appear  both  in  the 
denominators  and  in  the  numerators  or  integral  terms ;  and  also 
from  negative  and  fractional  exponents. 

184.  A  Homogeneous  Equation  is  one  in  which  the  sura 
of  the  exponents  of  the  unknown  quantities  in  each  term 
containing  unknown  quantities  is  the  same.     Thus, 

4:X^  —  4:xy  +/  — 16 
or  a;«+  Sxf+Bx'y  +  y'  =  21 

or  X*  —  4:X^y  +  Gx^/  _  43,^8  _|_  ^4  _  256 

is  a  homogeneous  equation. 

185.  Two  quantities  enter  Symmetrically  into  an  equa- 
tion when,  whatever  their  values,  they  can  exchange  places 
without  destroying  the  equation.     Thus, 

a:-  — 2a:y  +  /  r=  25 
or  a:»  +  3x2y4-3a:.y2  4-  /=    8 

or  x^'^2xy-\-y^+     2x  +  2y::=24: 


QUADRATIC   EQUATIONS.  197 

186.  Quadratic  equations  containing  two  unknown  quan- 
tities can  generally  be  solved  by  the  rules  already  given, 
if  they  come  under  one  of  the  three  following  cases  :  — 

I.    When  one  of  the  equations  is  simple  and  the  other 
quadratic. 

II.    When  the  unknown  quantities  enter  symmetrically 
into  each  equation. 

III.    When  each  equation  is  quadratic  and  homogeneous. 


CASE  I. 

V 

187.    When  one  of  the  equations  is  simple  and  the  other 
quadratic. 

1.   Given    1^^  +  ^^,'"    ^^l,  to  find  x  and  3^. 


OPERATION. 

2^:4-2^  =  22           (1)                               Sx'-{-7f=lll  (2) 

y=ll—x  (3)      3ar'-fl21  — 22a;  +  a;2=lll  (4) 

4a:-— 22a:  =  — 10  (5) 

42x2— ()-j- 112=  121—40  =  81  ^Q^ 

4a:=ll  ±9  =  20,  or  2  (7) 

y  =  6,orlO^(95                                        a:=5,  or^  (8) 

From  (1)  we  obtain  (3),  or  y  =  11  —  x.  Substituting  this  value 
of  y  in  (2),  we  obtain  (4),  an  affected  quadratic  equation,  which 
reduced  gives  (8) ;  and  substituting  these  values  of  x  in  (3),  we 
obtain  (9). 

In  this  Case  the  values  of  the  unknown  quantities  can 
generally  be  found  by  substituting  in  the  quadratic  equation 
the  value  of  one  unknown  quantity  found  by  reducing  the 
siviple  equation. 


198  ELEMENTARY  ALGEBRA. 

2.    Given   <       ^-^  f-  ,  to  find  x  and  y. 

^^  =  28       (1) 


OPERATION. 

ar  — y  =      3 

(2) 

V  — 2xy  +  2/'=      9 

(3) 

4a:2/            =112 

(4) 

x'  +  2xy  +  f=\2l 

(5) 

x  +  y  =±11 

(6) 

2x  =  14,  or—    8 

a) 

2y  =    8,  or—  14 

(8) 

X  =    7,  or  —    4 

(9) 

y  =    4,  or  —    7 

(10) 

Adding  four  times  (1)  to  the  square  of  (2),  we  obtain  (5)  ;  ex- 
tracting the  square  root  of  each  member  of  (5),  we  obtain  (6) ; 
adding  (2)  to  (6),  we  obtain  (7)  ;  subtracting  (2)  from  (6),  we  ob- 
tain (8)  ;  and  reducing  (7)  and  (8),  we  obtain  (9)  and  (10). 

Note.  —  Though  Example  2  can  be  solved  by  the  same  method 
as  Example  1,  the  method  given  is  preferable. 

By  this  method  find  the  values  of  x  and  y  in  the  fol- 
lowing equations ;  — 

3.  Given   j^  "2'  =    H-  Ans.   1^  =  6- 

4.  Given   j*+^=in.  Ans''  =  ^'»^«- 

ix^-\-fz=Sf>)  (^,  =  6,  or7. 

5.  Given   I  :^y  =  20) 

<  5  X  +  y  =  29  t 

6.  Given   \  x3,  =  24> 

iSx  —  2ij=  10) 


QUADRATIC  EQUATIONS.  199 

CASE    II. 
188.    When  the  unknown  quantities  enter  symmetrically 
into  each  equation. 

1.    Given    i    „    '   ^  f-  ,  to  find  x  and  y. 

I  :c3  +  /  r=  152  )  • 

OPERATION. 

a:+y=    8  (1)        x«  +  /  =  152     (2) 

x^  +  2xy  +  /  =  64  (3) 

(5) 
(6) 

a) 

(8) 


Sxy            =45 

xy            =15 

X-  - 

-2xy+y'=    4 

a:  — y  =  ±2 

2  a:  =  10,  or    6  (9) 

2y  =    6,  or  10  (10) 

X  z=    5,  or    3  (11) 

y  =    3,  or    5  (12) 

Squaring  (1),  we  obtain  (3);  dividing  (2)  by  (1),  we  obtain  (4); 
subtracting  (4)  from  (3),  we  obtain  (5),  from  which  we  obtain  (6) ; 
subtracting  (6)  from  (4),  we  obtain  (7)  ;  extracting  the  square  root 
of  each  member  of  (7),  we  obtain  (8)  ;  adding  (8)  to  (1),  we  ob- 
tain (9)  ;  subtracting  (8)  from  (1),  we  obtain  (10) ;  and  reducing 
(9)  and  (10),  we  obtain  (11)  and  (12). 

Note  1.  —  It  must  not  be  inferred  that  x  and  y  are  equal  to 
each  other  in  these  equations ;  for  when  x  ^  5,  y  =  3  ;  and  when 
ar  =  3,  y  ==  5.  In  all  the  equations  under  this  Case  the  values  of 
the  two  unknown  quantities  are  interchangeable. 

Note  2. — Although  af^ -\- y^  =  152  is  not  a  quadratic  equation, 
yet  as  we  can  combine  the  two  given  equations  in  such  a  manner 
as  to  produce  at  once  a  quadratic  equation,  we  introduce  it  here. 


200  ELEMENTARY  ALGEBRA. 

2.    Given    \        .  ^^  ^  ^  I ,  to  find  a:  and  y. 


x^  +  f  —  2x  —  2y 

OPERATION. 


xy^e    (1) 


x'-]-f—2x—2y=    3 

(2) 

2xy                        =12 

(3) 

(x-^yy-2(x^y)  =  15 

(4) 

(^  +  yy-  ()4-    1=16 

(5) 

x-\'y=l±4  =  5yOT  —  B 

(6) 

—  3±\/- 
x  =  3,or2,  or ^ 

-''a^ 

-SqpV^- 

-15 

y=2,or3,or— -^"^ (8) 

Adding  twice  (1)  to  (2),  we  obtain  (4) ;  completing  the  square 
in  (4),  we  obtain  (5) ;  extracting  the  square  root  of  each  member 
of  (5),  and  transposing,  we  obtain  (6) ;  and  combining  (1)  and  (6) 
as  the  sum  and  product  are  combined  in  the  preceding  example, 
we  obtain  (7)  and  (8). 

In  Case  II.  the  process  varies  as  the  given  equations 
vary.  In  general  the  equations  are  reduced  by  a  proper 
combination  of  the  sum  of  the  squares,  or  the  square  of  the 
sum  or  of  the  difference,  with  multiples  of  the  product  of 
the  two  unknown  quantities;  and  finally,  of  the  sum,  with 
the  difference  of  the  two  unknown  quantities. 

Note  S.  —  When  the  unknown  quantities  enter  into  each  equation 
symmetrically  in  all  respects  except  their  signs,  the  equations  can  be 
reduced  by  this  same  method ;  e.  g.  a:  —  ;/  =  7,  and  3^  —  ^  =  511. 
In  such  equations  the  values  of  the  unknown  quantities  are  not  inters 
changeable. 

Note  4.  —  The  signs  ±  q:  standing  before  any  quantity  taken  in- 
dependently are  equivalent  to  each  other ;  but  when  one  of  two  quan- 
tities is  equal  to  ±  a  while  the  other  is  equal  to  ^  ft,  the  meaning  is 
that  the  first  is  equal  to  -}-  a,  when  the  second  is  equal  to  —  h\  and 
the  first  to  —  a,  when  the  second  is  equal  to  -\-h. 


QUADRATIC   EQUATIONS.  201 

By  this  method  find  the  values  of  x  and  y  in  the  follow 
ing  equations :  — 

3.  Given    \^\-^^yr    ''I'       Ans.    j-^^^orS. 

4.  Given    \^  -  V  ^      H  ^^^     (a:  =  9,or-l. 

U3_/=z728i  \y=z  1,  or— .9. 

5.  Given   |^+^/^+y  =14) 
Note.  —  Divide  the  second  equation  by  the  first. 

6.  Given    1^  -V^  +  y  =      U. 

(x2+       xy^y'^lZZ) 

CASE    III. 
189.    When  each  equation  is  quadratic  and  homogeneous. 

1.   Given    -<     ^V    \     V  f-  >    to  find  x  and  y. 

XZx"   —xy—  10)  ^ 

OPERATION. 

2xy+y^5  (1)  3:r2  — a:y=10  (2) 

Let  X  =11  vy 
2vy'^-\-y'^  =  b  (3)       3  ^^3/2  _  ^^2  _  ^q  ^4^ 

2  y  -f  1  3  y2  —  y  '^ '  >' 

15  ^2  —  5  V  =  20  y  +  10         (8) 

3^2  _  5^ —  2  (9) 

t;  — 2,  or  — ^      (10) 

^  =  4^'    or^^^--     (11) 
2/=  ±  1,  or±-v/15         (12) 

.T  =  vy  =  ±  2,  or  q=  ^  \/T5      (13) 
9* 


202  ELEMENTARY   ALGEBRA. 

Substituting  vy  for  x  in  (1)  and  (2),  we  obtain  (3)  and  (4)  ;  from 
(3)  and  (4)  we  obtain  (5)  and  (6) ;  putting  these  two  values  of  y' 
equal  to  each  other,  we  obtain  (7),  which  reduced  gives  (10)  ;  sub- 
stituting this  value  of  v  in  (5),  we  obtain  (11),  which  reduced  gives 
(12)  ;  and  substituting  m  x  =  vy  the  values  of  v  and  y  from  (10)  and 
(12),  we  obtain  (13). 

Examples  under  Case  III.  can  generally  be  reduced  best 
by  substituting  for  one  of  the  unknown  quantities  the  product 
of  the  other  by  some  unknown  quantity,  and  then  finding  the 
value  of  this  third  unknown  quantity.  When  the  value  of 
this  third  quantity  becomes  known,  the  values  of  the  given 
unknown  quantities  can  be  readily  found  by  substitution. 

Note.  —  Whenever,  as  in  the  example  above,  the  square  root  is 
taken  twice,  each  unknown  quantity  has  four  values ;  but  these  values 
must  be  taken  in  the  same  order,  i.  e.  in  the  example  above,  when 
y  =  -}-l,  a;  =  -f-2;  when  y  =  —  l,a:  =  —  2;  when  y  =  -[-  \'~ib, 
x^=  —  i  y/  lo  ;  and  when  y  =  —  \/15,  a:  =  -j-^V/l5. 

By  this  method  find  the  values  of  x  and  y  in  the  follow- 
ing equations  :  — 


2.   Given    |     ^^-^y  =14) 


2y^ 

Ana.    <  

y=  ±  5,  or  ±  11  V  — i 

3.   Given    j-^+3-y  =  2T) 


(  V  = 


A„8.    f^=±3,or±9V-J. 
(j/z=  ±  2,  or  q=  8V— i- 

4.  Given    |x' +  4xy  =  U  -  2/) 

(.a;y  — 3/=    3  —  x'    )  

An«.    fx=±2,or±24V-j^. 
<y=  ±  l,or  T  H\/— iV- 

5.  Given    P2  ^  3x,v  =  2.^ -y' > 

(.        x"  — 3=y  +  2      ) 


QUADRATIC   EQUATIONS.  203 

190.    Find  the  values  of  x  and  y  in  the  following 

Examples. 

Note.  —  Some  of  the  examples  given  below  belong  at  the  same 
time  to  two  Cases.  Thus  in  Example  1  both  the  equations  are 
symmetrical,  and  both  are  quadratic  and  homogeneous,  and  there- 
fore it  belongs  both  to  Case  II.  and  Case  III.  Example  3  belongs 
both  to  Case  I.  and  Case  II. 


1.  Given   j   ^     ^y  =  ^n.      Ans.    I^  = 

2.  Given    \  ^-^^    ^  I. 

3.  Given    .^  ^  +^=    ^  I. 

U2+.y  =  32-(:r  +  y2)> 


=  ±  5,  or  ±  4. 
±  4,  or  ±  5. 


Ans.    5^  =  4' or  3. 
<^  =  3,  or  4. 


4.    Given    \        ^^  =  12  ) 

\x^-\-x  =  Z1—y  —  y'-S 

\       ^='^.  Ans.    j-  =  ^.or3. 


6.    Given 


6.    Given    | 


x-y=.        2) 
r:^/^  2^:5^  =  1295) 

Note.  —  Considering  a:y  a  single  quantity,  find  its  value  in  the 
second  equation. 

7.    Given    fx^y-xy^=30) 

Note.  —  Subtract  from  the  second  equation  three  times  the  first, 
and  extract  the  cube  root  of  each  member  of  the  resulting  equation. 


8.  Given    i  2^^  +  2x/ =  168| . 
(  a;»  4-  v«  =    91  ) 


A  ns.     -l 

w  =  3,  or  4 


204  ELEMENTARY  ALGEBRA. 

9.   Given    5  3a-^  -  3./  =  18) 

10.  Given    \^        ^  ~  3  ^  • 

(  a:y=10  ) 

An8.    |^=±5. 

11.  Given    |^-^-2.y  =  88> 

Ans     f^=±4,  or±    66  V^. 
(2^=  ±  3,  or  q:n5Vij(^5. 

12.  Given    f  x^  -  2x  +  2^/ =  30 -y^ ) 

(  4ic2/=z60  > 

13.  Given    i      ^    -V^^/) 

Ans.    1^-1000,  or  8. 
(y=    625,  or  1. 

1.     n-  f        3;ryiiiil8) 

14.  Given    \  ^  v  - 

(a:^  — y  =  65i  

.  (ar==  ±  3,  or  ±  2>C^  —  1. 

Ans.     <  -1-      ;  -1-       "V 

(y  =  ±  2,  or  ±  3/i/— 1. 

15.  Given    |3  (x  -  y)  ===  3  (^x  +  ^Z^)  ) 

(  a:yr=36  ) 

X  =  — ,  or  9,  or  4. 

y  =  — ^^ ^ ,  or  4,  or  9. 

16.  Given   V^"  V^  = -T""*  I. 

17.  Given    |*   ^*   —    '|. 

(x    —  y   =19) 


QUADRATIC   EQUATIONS.  205 

18.  Given    -l^      -r  -^  if  c  . 

19.  Given    |  ^~' —  ^"' =^  ^"^  I . 

20    Given    j  ^' +  2:^'y  +  2a:y2  _^  /  =  95> 


21.   Given    I         ^^  ==      H  . 
(a;4  +  3/4  =  272) 


22.   Given    |^  +.^/  =      H 


PROBLEMS 

PRODUCING   QUADRATIC  EQUATIONS    CONTAINING   TWO 
UNKNOWN   QUANTITIES. 

191.  Though  the  numerical  negative  values  obtained 
in  solving  the  following  Problems  satisfy  the  equations 
formed  in  accordance  with  the  given  conditions,  they  are 
practically  inadmissible,  and,  except  in  Example  4,  are 
not  given  in  the  answers. 

1.  The  sum  of  the  squares  of  two  numbers  plus  the 
sum  of  the  two  numbers  is  98  ;  and  the  product  of  the 
two  numbers  is  42.     Wh^t  are  the  numbers  ? 

Ans.   7  and  6. 

2.  If  a  certain  number  is  divided  by  the  product  of  its 
figures  the  quotient  will  be  3  ;  and  if  18  is  added  to  the 
number,  the  order  of  the  figures  will  be  inverted.  What 
is  the  number  ^  Ans.  24. 

3.  A  certain  number  consists  of  two  figures  whose 
product  is  21  ;   and  if  22  is  subtracted  from   the  number. 


206  ELEMENTARY   ALGEBRA. 

and  the  sum  of  the  squares  of  its  figures  added  to  the 
remainder,  the  order  of  the  figures  will  be  inverted.  What 
is  the  number?  Ans.  37. 

4.  Find  two  numbers  such  that  their  sum,  their  prod- 
uct, and  the  difference  of  their  squares  shall  be  equal  to 
one  another.  Ans.  f  ±  ^  V5  and  ^  ±  ^>v/5. 

6.  There  are  two  pieces  of  cloth  of  different  lengths ; 
and  the  sum  of  the  squares  of  the  number  of  yards  in 
each  is  145  ;  and  one  half  the  product  of  their  lengths 
plus  the  square  of  the  length  of  the  shorter  is  100.  What 
is  the  length  of  each  ? 

Ans.  Shorter,  8  ;  longer,  9  yards. 

6.  Find  two  numbers  such  that  the  greater  shall  be  to 
the  less  as  the  less  is  to  2f,  and  the  difference  of  their 
squares  shall  be  33. 

7.  The  area  of  a  rectangular  field  is  1575  square  rods ; 
and  if  the  length  and  breadth  were  each  lessened  6  rods, 
its  area  would  be  1200  square  rods.  What  are  the  length 
and  breadth  ? 

8.  Find  two  numbers  such  that  their  sum  shall  be  to 
6  as  9  is  to  the  greater,  and  the  sum  of  their  squares 
shall  be  45.  Ans.  9  VT  and  3  \^'^,  or  6  and  3. 

9..  The  fore  wheels  of  a  carriage  make  2  revolutions 
more  than  the  hind  wheels  in  going  90  yards  ;  but  if  the 
circumference  of  each  wheel  is  increased  3  feet,  the  car- 
riage must  pass  over  132  yards  in  order  that  the  fore 
wheels  may  make  2  revolutions  more  than  the  hind  wheels. 
What  is  the  circumference  of  each  wheel  ? 

Ans.  Fore  wheels,  13J  feet;  hind  wheels,  15  feet. 

10.  Find  two  numbers  such  that  five  times  the  square 
of  the  greater  plus  three  times  their  product  shall  be  104, 
and  three  times  the  square  of  the  less  minus  their  prod- 
uct shall  be  4. 


KATIO   AND   PEOPORTION.  207 

SECTION    XXT. 

RATIO    AND    PROPORTION. 

192.  Ratio  is  the  relation  of  one  quantity  to  another  of 
the  same  kind  ;  or,  it  is  the  quotient  which  arises  from 
dividing  one  quantity  by  another  of  the  same  kind. 

Ratio  is  indicated  by  writing  the  two  quantities  after 
one  another  with  two  dots  between,  or  by  expressing  the 
division  in  the  form  of  a  fraction.  Thus,  the  ratio  of  a  to 
h  is  written,  a  :  5,  or  r  ;  read,  a  is  to  h^  or  a  divided  by  h. 

193.  The  Terms  of  a  ratio  are  the  quantities  compared, 
whether  simple  or  compound. 

The  first  term  of  a  ratio  is  called  the  antecedent,  and  the 
other  the  consequent ;  and  the  two  terms  together  are  called 
a  couplet. 

194.  An  Inverse,  or  Reciprocal  Ratio,  of  any  two  quan- 
tities is  the  ratio  of  their  reciprocals.     Thus,  the  direct  ratio 

of  a  to  b  \s  a  :  b,   i.  e.  r;   and  the  inverse  ratio  of  a  to  i  is 

1     1     .        1        1        &  , 

-  :  7>   1.  e.  -  -i-  7  =  -'    or  6  :  a. 
a     0  aba 

195.  Proportion  is  an  equality  of  ratios.  Four  quan- 
tities are  proportional  when  the  ratio  of  the  first  to  the 
second  is  equal  to  the  ratio  of  the  third  to  the  fourth. 

The  equality  of  two  ratios  is  indicated  by  the  sign 
of  equality  (==)  or  by  four  dots  (:  :). 

Thus,  a  :  b  =  c  :  d,  or  a  :  b :  :  c  :  c?,  or  t  =  -, ;  read,  a  to  b 

b       d 

equals    c  to  d,  or  a  is  to  J  as  c  is  to  d,  or  a  divided  by  b 
equals  c  divided  by  d. 


208  ELEMENTARY   ALGEBRA. 

196.  In  a  proportion  the  antecedents  and  consequents 
of  the  two  ratios  are  respectively  the  antecedents  and  con- 
sequents of  the  proportion.  The  first  and  fourth  terms  are 
called  the  extremes,  and  the  second  and  third  the  means. 

107.  When  three  quantities  are  in  proportion,  e.  g. 
a  :  b  =z  b  :  c,  the  second  is  called  a  mean  proportional  be- 
tween the  other  two  ;  and  the  third,  a  third  proportional 
to  the  first  and  second. 

198.  A  proportion  is  transformed  by  Alternation  when 
antecedent  is  compared  with  antecedent,  and  consequent 
with  consequent. 

199.  A  proportion  is  transformed  by  Inversion  when 
the  antecedents  are  made  consequents,  and  the  conse- 
quents antecedents. 

200.  A  proportion  is  transformed  by  Composition  when 
in  each  couplet  the  sum  of  the  antecedent  and  consequent 
is  compared  with  the  antecedent  or  with  the  consequent. 

201.  A  proportion  is  transformed  by  Division  when  in 
each  couplet  the  difference  of  the  antecedent  and  conse- 
quent is  compared  with  the  antecedent  or  with  the  con- 
sequent. 

THEOREM    I. 

202.  In  a  proportion  the  product  of  the  extremes  is  equal 
to  the  product  of  the  means. 

Let  a  :  bz=z  c  :  d 

a        c 

Clearing  of  fractions,  ad  =  be 


RATIO  AND   PROPORTION.  209 

THEOREM    II. 
203t   If  the  product  of  tvjo  quantities  is  equal  to  the  prod- 
uct of  two  others,  the  factors  of  either  product  may  be  made 
the  extremes,  and  the  factors  of  the  other  the  means  of  a 
proportion. 

Let  ad  =  be 

Q  C 

Dividing  hj  bd,  f  ^^  ^ 

i.  e.  a  :  b  =  c  :  d 

THEOREM    III. 

204.  If  four  quantities  are  in  proportion,  they  will  be  in 
proportion  by  alternation. 

Let  a  :  b  =  c  :  d 

By  Theorem  I.  ad  =  be 

By  Theorem  II.  a  :  c  =  b  :  d 

THEOREM    IV. 

205.  If  four  quantities  are  in  proportion,  tJiey  will  be  in 
proportion  by  inversion. 

Let  a  :  b  =  c  :  d 

By  Theorem  I.  ad^^bc 

By  Theorem  II.  b  :  a  =  d  :  c 

THEOREM    V. 

206.  If  three  quantities  are  in  proportion,  the  product  of 
the  extremes  is  equal  to  the  square  of  the  mean. 

Let  a  :  b  =  b  :  c 

By  Theorem  I^  _       ac  =  b"^ 

THEOREM    VI. 

207.  If  four  quantities  are  in  proportion,  they  will  be  in 
proportion  by  composition. 


ELEMENTARY 

ALGEBRA. 

a'.b  = 

:c:d 

a 
b  — 

c 

d 

lember, 

l  +  ^  = 

1  +  ^ 

b      ~ 

c-f-rf 
d 

a 

+  b:b  = 

c  +  d 

210 
Let 
i,  e. 

Addi 
or 
i.  e. 


THEOREM    VII. 

208.   If  four  quantities  are  in  proportion,  they  will  be  in 
proportion  by  division. 

Let  a  :  b  =  c  :  d 

a        c 

i  =  i 

Subtracting  I  from  each  member,  v  —  I  =^  ~  —  1 

a—hc—d 

i.  e.  a  —  b  :  b=z  c  —  did 

THEOREM    VIII. 
^09.    Two  ratios  respectively  equal  to   a  third  are  equil 


to  each  other. 


Let  a  '.  b  =z  m  '.  n  and  c  :  d  =  m  :  n 

.      '  a        m  .  c        m 

1.  e.  T  =  -         and         ^  =  - 

on  d        n 


Hence  (Art.  13,  Ax.  8), 

i.  e.  a  :  b=zc  :  d 


a c 

b~d 


THEOREM    IX. 
210.   If  four  quantifies  are  in  proportion,  the  sum   and 
difference  of  the  terms  of  each  couplet  will  be  in  proportion. 


RATIO   AND   PROPORTION.  211 

Let  a  :  h  =z  c  '.  d 

By  Theorem  VI.  a -\- b  :  h  =  c -\- d  :  d    {I) 

and  by  Theorem  VII.  a  — b.b  =  c  — did    (2) 

From  (1),  by  Theorem  III.  a-{-b:  c-\-d=b:  d 

From  (2),  by  Theorem  III.  a  —  b:c  —  d=b'.d 

By  Theorem  VIII.  a-\-b:  c-\-d=a  — bic  —  d 

Hence,  by  Theorem  III.  a  -\-b  :  a  —  b  =  c  -\-  d  :  c  —  d 

THEOREM    X. 

211.  Equimultiples  of  two  quantities  ham  the  same  ratio 
as  the  quantities  themselves. 

For  by  Art.  83,  ?  =  ^ 

•^  '  h        mb 

i.  e.  a  :  b  =:  ma  :  mb 

Cor.  It  follows  that  either  couplet  of  a  proportion  may 
be  multiplied  or  divided  by  any  quantity,  and  the  result- 
ing quantities  will  be  in  proportion.  And  since  by  Theo- 
rem III.  if  a  :  b  =  ma  :  mb,  a  :  ma  =  b  :  mb,  or  ma  :  a 
=z  mb  :  b,  it  follows  that  both  consequents,  or  both  ante- 
cedents, may  be  multiplied  or  divided  by  any  quantity, 
and  the  resulting  quantities  will  be  in  proportion. 

THEOREM    XI 

212.  If  four  quantities  are  in  proportion,  like  powers  or 
like  roots  of  these  quantities  will  be  in  proportion. 

Let  a  :  b  =  c  :  d 

i.  e. 

Hence, 

i.  e.  a""  :  b""  =^  c""  :  d^ 

Since  n  may  be  either  integral  or  fractional,  the  theorem 
is  proved. 


a 
b 

= 

c 
d 

6« 

== 

d^ 

212  ELEMENTARY   ALGEBRA. 

THEOREM    XII. 
213.   If  any  number  of  quantities  are  proportional,    any 
antecedent  is  to  its  consequent  as  the  sum  of  all  the  antece- 
dents is  to  the  sum  of  all  the  consequents. 

Let  a  :  b  =  c  :  d=:  e  :f 

Now  ab=:ab  (1) 

and  by  Theorem  I.  ad  =  bc  (2) 

and  also  af=:be  (3) 

Adding(l),(2),(3),    a{b  +  d+f)=b{a-\-c-^e) 
Hence,  by  Theorem  II.  a:b=za-\-c-{-e'.b-\-d -\-f 

THEOREM    XIII. 
21 4 1   If  there  are  two  sets  of  quantities  in  proportion,  their 
products,  or  quotients,  term  by  term,  will  be  in  proportion. 


Let 

a  :  b=z  c  :  d 

and 

e:f=g  :h 

By  Theorem  I. 

ad  =z  be 

(1) 

and 

eh=fg 

(2) 

Multiplying  (1)  by  (2), 

ad  eh  ^  b  efg  ; 

(3) 

Dividing  (1)  by  (2),  . 

ad       be 

(4) 

From  (3),  by  Theorem  II. 

a  e  :  bf  =^  e  g  :  dh 

and  from  (4), 

a     b        c     d 

PROBLEMS    IN    PROPORTION. 

215.  By  means  of  the  principles  just  demonstrated,  a 
proportion  may  often  be  very  much  simplified  before 
making  the  product  of  the  means  equal  to  the  product 
of  the  extremes  ;  and  a  proportion  which  could  not  oth- 
erwise be  reduced  by  the  ordinary  rules  of  Algebra  may 
often  be  so  simplified  as  to  produce  a  simple  equation. 


RATIO   AND   PROPORTION.  213 

1.  The  cube  of  the  smaller  of  two  numbers  multiplied 
by  four  times  the  greater  is  96  ;  and  the  sum  of  their 
cubes  is  to  the  difference  of  their  cubes  as  210  :  114. 
What  are  the  numbers  ? 

SOLUTION. 

Let  X  =  the  greater  and  y  =.  the  less. 

Then     4a;/  =  96    (1)         a^^  +  ^/^a:^  — /  ==  210  :  114    (2) 
From  (2),  by  Theo.  X.,  Cor.   x^ -\-y^ :  x^ —  y^  =  Zb  :  19 
By  Theorem  IX.  2a;«  :  2/  =  54  :  16 

By  Theorem  X.,  Cor.  x^  '.y^  =  21  :^ 

By  Theorem  XI.  x:y  =  Z:2 

By  Theorem  I.  2  ;r  =  3  y  (3) 

From  (1)  and  (3)  we  find  a:  =  3  and  y  =  2. 

2.  The  product  of  two  numbers  is  V8  ;  and  the  differ- 
ence of  their  cubes  is  to  the  cube  of  their  difference  as 
283  :  49.     What  are  the  numbers  ? 

SOLUTION. 

Let       X  =  the  greater  and  y  =  the  less. 

Thena;2/=78    (1)     3^  —  f :  xr^ —  3  sfiy -{- 3xf —  f  =  283  :49  (2) 
From  (2),  by  division,  3x'y  —  3xy'^  :  (x  —  yY  =  234  :  49 

Dividing  1st  couplet  by  a:  —  y,  3xy  :  (x  —  yY  =  234  :  49 

Dividing  antecedents  by  3,  xy  :  (x  —  yY  =  78  :  49 

Substituting  the  value  of  xy,  78  :  (a:  —  yy  =  78  :  49 

•Dividing  antecedents  by  78,  1  :  (x  —  yy  =  1  :4d 

Extracting  the  square  root,  1  :  x  —  y  =  l  :  7 

Whence,  x  —  y  ==  7  (3) 

From  (1)  and  (3)  we  find  a:  =  13  and  y  =  6. 

3.  The  sum  of  the  cubes  of  two  numbers  is  to  the  cube 
of  their  sum  as  13  :  25  ;  and  4  is  a  mean  proportional  be- 
tween them.     What  are  the  numbers? 


214  ELKMENTARY   ALGEBRA. 

4  The  difference  of  two  numbers  is  10  ;  and  their  prod- 
uct is  to  the  sum  of  their  squares  as  6  :  37.  What  are 
the  numbers  ? 

SOLUTION. 

Let                     X  =  the  greater  and  y  =  the  less. 

Then           x—y=10         (1)  ary  :  r» -|- ^  =    6  :  37   (2) 

From  (2),  by  Theorem  X.,  Cor.  2zy::c2-}-y^=12:37 
By  Theorem  IX.         x"  -\-  2  x y  -\-  f  :  x""  —  2  xy  -{-  f  =  40  :  2o 

By  Theorem  XL  x  -J[-  y  :  x  —  y=    7:5 

By  Theorem  IX.  2  i:  :  2  y  =  12  :  2 

By  Theorem  X.,  Cor.  x  :  y  =6:1 

By  Theorem  I.  x  ^  6  y          (S) 

From  (1)  and  (3)  we  find  a:  =  12  and  y  =  2. 

5.  The  product  of  two  numbers  is  136  ;  and  the  dif- 
ference of  their  squares  is  to  the  square  of  their  differ- 
ence as  25  :  9.     What  are  the  numbers  ?     Ans.  8  and  17. 

6.  As  two  boys  were  talking  of  their  ages,  they  dis- 
covered that  the  product  of  the  numbers  representing 
their  ages  in  years  was  320,  and  the  sum  of  the  cubes 
of  these  same  numbers  was  to  the  cube  of  their  sum  as 
7  :  27.     What  was  the  age  of  each  ? 

Ans.  Younger,  16;  elder,  20  years. 

7.  As  two   companies   of  soldiers   were  returning  from 
the  war,  it  was  found  that  the  number  in  the  first  multi- 
plied by  that  in  the  second  was  486,  and  the  sum  of  the  , 
squares  of  their  numbers  was  to  the  square  of  the  sum  as 
13  :  25.     How  many  soldiers  were  there  in  each  company? 

Ans.  In  1st.  27;  in  2d,  18. 

8.  The  difference  of  two  numbers  is  to  the  less  as  100 
is  to  the  greater ;  and  the  same  difference  is  to  the  greater 
as  4  is  to  the  less.     What  are  the  numbers  ? 

Note.  —  Multiply  the  two  proportions  together.     (Theorem  XIIL) 


PROGRESSION.  215 


SECTION   XXII. 

PROGRESSION. 

216.  A  Progression  is  a  series  in  which  the  terms  in- 
crease or  decrease  according  to  some  fixed  law. 

217.  The  Terms  of  a  series  are  the  several  quantities, 
whether  simple  or  compound,  that  form  the  series.  The 
first  and  last  terms  are  called  the  extremes^  and  the  others 
the  means. 

ARITHMETICAL    PROGRESSION. 

218.  An  Arithmetical  Progression  is  a  series  in  which 
each  term,  except  the  first,  is  derived  from  the  preced- 
ing by  the  addition  of  a  constant  quantity  called  the  com- 
mon difference. 

219.  When  the  common  difference  is  positive,  the  series 
is  called  an  ascending  series,  or  an  ascending  progression  ; 
when  the  common  difference  is  negative,  a  descending  se- 
ries.    Thus, 

a,  a  -\-  d,  a  -\-  2d,  a  -\-  '6d,  &c. 

is  an  ascending  arithmetical   series  in  which  the  common 
difierence  is  d ;  and 

a,  a  —  d,  a  —  2  c?,  a  —  3  c?,  &c. 

is  a  descending  arithmetical  series  in  which  the  common 
difference  is  —  d. 

220.  In  Arithmetical  Progression  there  are  five  elements, 
any  three  of  which  being  given,  the  other  two  can  be 
found  :  — 

1.  The  first  term. 

2.  The  last  term. 


216  ELEMENTARY   ALGEBRA. 

3.  The  common  difference. 

4.  The  number  of  terms. 

6.    The  sum  of  all  the  terms. 

221.  Twenty  cases  may  arise  in  Arithmetical  Progres- 
sion.    In  discussing  this  subject  we  shall  let 

a  =  the  first  term, 

/  =  the  last  term, 

d  =  the  common  difference, 

n  =  the  number  of  terms, 

aS^=:  the  sum  of  all  the  terms. 

CASE    I. 

222.  The  first  term,  common  difference,  and  number  of 

terms  given,  to  find  the  last  term. 

In  this  Case  a,  d,  and  n  are  given,  and  I  is  required.     The  suc- 
cessive terms  of  the  series  are 

a,  a-\-d,  a  -f  2  J,  a-\-  3d,  a  -\-  Ad,  &c. ; 

that  is,  the  coefficient  of  d  in  each  term  is  one  less  than  the  number 
of  that  term,  counting  from  the  left ;  therefore  the  last  or  nth  term  in 
the  series  is 

a  -}-  (n  —  1)  d 

or  1  =  a  -j-  {n  —  1)  d 

in  which  the  series  is  ascending  or  descending  according  as  d  is  posi- 
tive or  negative.     Hence, 

RULE. 

To  the  first  term  add  the  product  formed  by  multiplying  the 
common  difference  by  the  number  of  terms  less  one. 

1.  Given  a  =  4,  d  =:2,  and  w  =  9,  to  find  /. 

l  =  a-\-{n  —  l)  rf  =  4  +  (9  — 1)  2  =  20,  Ans. 

2.  Given  a  =  T,  d=3,  and  n  =  19,  to  find  L 

Ans.  /=61. 


PROGRESSION.  217 

3.  Given  a  =  29,  d  =  —  2,  and  n  =  14,  to  find  /. 

Ans.  1  =  ^. 

4.  Given  a  =  4.0,  d=  10,  and  n  =  100,  to  find  /. 

5.  Given  a=  I,  d  =:  ^,  and  n  =  17,  to  find  I. 

6.  Given  a  z=  ^,  d  =  —  J^,  and  w  =  13,  to  find  I. 

1.    Given  a=  .01,   d  =  —  .001,  and  n=  10,  to  find  /. 

CASE    II. 
223.   The  extremes  and  the  number  of  terms  given,  to 
find  the  sum  of  the  series. 

In  this  Case  a,  I,  and  n  are  given,  and  S  is  required. 

Now  S  =  a  4-  (a  +  rf)  +  (a  +  2rf)  +  (a  +  3rf)  + +  / 

or,  inverting  the  series,      S=  Z  +  (  Z  —  rf)  +  ( /  —  2rf)  +  (Z  — 3d)  4- +a 

Adding  these  together,  2  S=  (a  +  0  +  (a  +  Z)  +  (a  +  Z)  +  (a  +  0  + +  (a  +  Z) 

And  since  (a  -{-  Z)  is  to  be  taken  as  many  times  as  there  are  terms, 
hence  2S  =  n{a-{-l) 

or  -S?  =  9  («  -j-  0-     Hence, 

RULE. 

Find  one  half  the  product  of  the  sum  of  the  extremes  and 
the  number  of  terms. 

Note.  —  If  in  place  of  the  last  term  the  common  difference  is 
given,  the  last  term  must  first  be  found  by  the  Rule  in  Case  I. 

1.    Given  a  =  S,  1=  141,  and  n  =  26,  to  find  S. 

S=l(a-irl)  =  ^{S+  141)  =  1872,  Ans. 

^.    Given  a  =  i,  1=25,  and  n  =  63,  to  find  S. 

Ans.  S=19Si. 

3.  Given  a  =  4,  d  =z2,  and  n  =  24,  to  find  S. 

Ans.  5=648. 

4.  Given  a  ==  —  3,  c?  =  2,  and  w  =  4,  to  find  S. 

Ans.   S=0. 


218  ELEMENTARY   ALGEBRA. 

5.  Given  a  =  ^,  d  =  —  ^,  and  n  =    3,  to  find  S. 

6.  Given  a  =  .07,  /=  .11;  and  n=n,  to  find  S. 
1.    Given  a=  —  4^,  </  =  f,  and  n  =  25,  to  find  S. 

CASE   III. 
224.   The  extremes  and  number  of  terms  given,  to  find 
the  common  difference. 

In  this  case  a,  /,  and  n  are  given,  and  d  is  required. 
From  Case  I.  we  have  /  =  a  -|-  (n  —  1)  rf 

Transposing  and  reducing,  d  = .     Hence, 

BULE. 

Divide  the  last  term  minus  the  first  term  by  the  number 
of  terms  less  one,  and  the  quotient  will  be  the  common  dif- 
ference. 

1.  Given  a  =  6,  /  =  47,  and  n  =  1,  to  find  d. 

J         I  —  a         47  —  5         ^       . 

2.  Given  a  =  27,  /=  148,  and  n  =  12,  to  find  d. 

Ans.  d  =z  11. 

3.  Given  a  =  41,  Z=3,  and  n  =:  20,  to  find  d. 

Ans.  d  =  —  2. 

4.  Given  a  =z  ^l,  I  ^  j\,  and  n  =:  6,  to  find  d. 

Ans.  rf  =  —  »^. 

^.    Given  a  =  .09,  /=  .9,  and  n  =  10,  to  find  d. 

Note.  —  This  rule  enables  us  to  insert  any  number  of  arithmet- 
ical means  between  two  j.-iven  quantities ;  for  the  number  of  terms 
is  two  greater  than  the  number  of  means.  Hence,  if  m  =  the  num- 
ber of  means,  m-}-2  =  n,  or  m  4-  I  =  n  —  1,  and  d  =  — r-r* 

m-|-  1 

Having  found  the  common  difference,  the  means  are  found  by  add- 
ing the  common  difference  once,  twice,  &c.,  to  the  first  term. 


PROGRESSION.  219 

6.    Find  6  arithmetical  means  between  3  and  38. 

Ans.  8,  13,  18,  23,  28,  33. 

1.    Find  3  arithmetical  means  between  3  and  27. 

8.  Find  5  arithmetical  means  between  1  and  3t. 

9.  Find  7  arithmetical  means  between  2  and  26. 

Note.  —  When  m  =  1,  the  formula  becomes 

I  —  a 


Adding  a  to  each  member, 


''=      2 


2        '  2 

But  a  -]-  d  \s  the  second  term  of  a  series  whose  first  term  is  a  and 
common  difference  d,  or  the  arithmetical  mean  of  the  series  a,  a -\-  d, 
a  -}-  2d.  Hence,  the  arithmetical  mean  bettveen  two  quantities  is  one 
half  of  their  sum. 

10.  Find  the  arithmetical  mean  between  7  and  17. 

Ans.  12. 

11.  Find  the  arithmetical  mean  between  ^  and  J. 

12.  Find  the  arithmetical  mean  between  4  and  —  4. 

225.  From  the  formulas  established  iu  Arts.  222  and 
223,  viz. 

l^a  +  {n  —  l')d  (1) 

S=l{a  +  l)  (2) 

can  be  derived  formulas  for  all  the  Cases  in  Arithmetical 
Progression. 

From  (1)  we  can  obtain  the  value  of  any  one  of  the  four  quanti- 
ties, /,  a,  n,  or  d,  when  the  other  three  are  given ;  and  from  (2) 
the  value  of  any  one  of  the  four  quantities,  S,  n,  a,  or  Z,  when  the 
other  three  are  given.  Formulas  for  the  remaining  twelve  Cases 
which  may  arise  are  derived  by  combining  the  two  formulas  (1) 
and  (2),  so  as  to  eUminate  that  one  of  the  two  unknown  quantities 
whose  value  is  not  sought. 


220  ELEMENTARY    ALGEBRA. 

1.    Find  the  formula  for  the  value  of  n,  when  a,  d,  and 
S  are  given. 

OPERATION. 

I    =,a  +  {n-l)d  (1)     .  S=l{aJrl)      (2) 

ln=zan-\-dn^  —  dn  (3)     2S—an=ln  (4) 

an -\- dn^  —  dn  =  2S — an  (5) 

«'-(^)«=¥  («) 


»  = 2rf  W 

To  obtain  the  formula  required  in  this  example,  /  must  be  elim- 
inated from  (1)  and  (2).  From  (1)  and  (2)  we  obtain  (3)  and  (4). 
Placing  these  two  values  of  Zn  equal  to  each  other,  we  form  (5), 
which  reduced  gives  (8),  or  the  value  of  n  in  known  quantities. 

2.  Find  the  formula  for  the  value  of  n,  when  d,  /,   and 

S  are  given.  Ar.«,   .  -  2  /  +  ^/ ±  y- (27  4-~^)^  -  «  ^ '^ 

Ans.  n  —  -^ 

3.  Find  the  formula  for  the  value  of  5,  when  a,  d,  and 
n  are  given.  Ans.   5  =  | n  [2 a  +  (w  —  1)  </]. 

4.  Find  the  formula  for  the  value  of  aS',  when  a,  d,  and 
/  are  given.  j^^^    ^  _  (/  -|-  g)  (/  —  g  -|-  </) 

2  d 

5.  Find  the  formula  for  the  value  of  S,  when  d,  n,  and 
I  are  given.  Ans.   S  =  \  n  [11  —  {n  —  \)  d]. 

6.  Find  the  formula  for  the  value  of  /,  when  a,  d,  and 


S  are  given.  d         fl         d^ 


Ans.  ;  =  -|±y/(«-0   +2rf5. 


7.    Find  the  formula  for  the  value  of  /,  when  d,  n,  and 
*S  are  given.  .    g   j  _.^  S -{- n  (n  — i)  d ^ 

2n 


PROGRESSION.  221 

8.    Find  the  formula  for  the  value  of  d,  when  a,  n,  and 

♦S' are  sriven,  .  -,       2*5 — 2 an 

°  Ans.  d=  — -r-. 

n  (n  —  I) 

9.    Find  the  formula  for  the  value  of  d,  when  a,  /,  and 
S  are  given.  .  ,_  (I  -\- a)  Q  —  a) 

10.  Find  the  formula  for  the  value  of  d,  when  n,  I,  and 

5  are  given.  Ans.  e?=  '  ^"' "  ^>. 

n  (71  —  1) 

11.  Find  the  formula  for  the  value  of  a,  when  d,  n,  and 

S  are  given.  .  2 S  —  n(n--V)d 

Ans.    ct  —  — 

2n 

12.  Find  the  formula  for  the  value  of  a,  when  d,  I,  and 


S  are  given.  ^ns.  a  =  ^  ±  y/  (-^  +  ?)'  -  2  rf  & 

226.   To   find  any  one  of  the  five  elements  viheo.  three 
others  are  given. 

RULE. 
Substitute  the  given  values  in  that  formula  whose  first  mem- 
ber is  the  required  term,  and  whose  second  contains  the  three 
given  terms. 

1.    Given  d=:2,  /  =  *21,  and  S=  120,  to  find  a. 

OPERATION. 

d 
^=2 


±v/G+0-'^^^         ^^) 


l^jn 


y/(^  +  2l)   -2.2.120  (2) 

3,  or  —  1  (3) 


In  Example  12,  Art.  225,  we  find  (1),  the  required  formula;  substi- 
tuting the  given  values  of  </,  I,  and  S,  we  obtain  (2),  which  reduced 
gives  (3),  or  rt  =  3,  or  —  1. 

Note.— If  a  =  3,  n=  10;  but  if  a  =  —  1,  n=  12. 


222  ELEMENTARY  ALGEBRA. 

2.  Given  d=^,  .1=21,  and  ^S'^:  392,  to  find  n. 

Ana.  n  =  147,  or  16. 
Note.  —  When  n  =  147,  a  ==  —  21 1 ;  but  when  n  =  IG,  a  =  22 

3.  Given  d=1,  n  =  6,  and  S=  135,  to  find  l. 

Ans.  /  =  40. 

4.  Given  d  ==  —  2,  n  =  6,  and  a  =  5,  to  ^nd  /S'. 

Ans.  S=0. 

6.    Given  a  =  i,  ^=15,  and  S=2SS^,  to  find  rf. 

Ans.  rf  =  f . 

6.  Find  the  100th  term  of  the  series  3,  10,  17,  &c. 

Ans.  696. 

7.  Find  the  sum  of  100  terms  of  the  series  3,  10,  17,  &c. 

Ans.  34950. 

8.  Find  the  common  difference  and  sum  of  the  series 
whose  first  term  is  25,  last  term  95,  and  number  of 
terms  15.'  Ans.  d  =  5 ',  aS'=:900. 

9.  Find  the  sum  of  the  natural  series  of  numbers  from 
1  to  100,  inclusive. 

10.  Find  the  sum  of  10  of  the  odd  numbers  1,  3,  5,  &c. 

11.  Find  the  sum  of  10  of  the  even  numbers  2,  4,  6,  &c. 

12.  How  many  strokes  does  a  clock  strike  in  12  hours? 

13.  If  100  trees  stand  in  a  straight  line  10  feet  from 
one  another,  how  far  must  a  person,  starting  from  the 
first  tree  and  returning  to  it  each  time,  travel  to  go  to 
every  tree?  Ans.  18J  miles. 

14.  If  a  person  should  save  a  cent  the  first  day,  two 
cents  the  second,  three  the  third,  and  so  on,  how  much 
would  he  save  in  365  days?  Ans.  S667.95. 

15.  If  a  person  should  save  $25  a  year  and  put  this 
sum  at  simple  interest  at  6  per  cent  at  the  end  of  each 
year,  to  how  much  would  it  amount  at  the  end  of  25 
years  ? 


PROGRESSION.  223 

PROBLEMS 
TO  WHICH  THE  FORMULAS   DO  NOT  DIRECTLY  APPLY. 

227.  Sometimes  in  examples  in  progression  the  terms 
are  not  directly  given,  but  are  implied  in  the  conditions  of 
the  problem.  In  this  case  the  formulas  cannot  be  directly 
used,  but  the  terms  can  be  represented  by  unknown  quan- 
tities, and  equations  formed  according  to  the  given  con- 
ditions. 

228.  If  a;  =  first  term  and  y  =  the  common  diflference ; 
then 

X,  x-\-y,  x-\-ty,  x-\-  3y,  &c. 

will  represent  the  series. 

It  will  often  be  found  more  convenient  when  the  num- 
ber of  terms  is  odd  to  represent  the  middle  term  by  x 
and  the  common  difference  by  y ;  then  the  series  for  three 
terms  will  be 

x  —  y,  X,  x-\-y] 
and  for  five  terms, 

x  —  2y,  x—y,  x,  x-\-y,  x-\-2y; 

and  when  the  number  of  terms  is  even,  to  represent  the 
two  middle  terms  by  cc  —  y  and  x  -\-  y,  and  the  common 
difference  by  2  y ;  then  the  series  for  four  terms  is 

x  —  Sy,     x  —  y,    x  +  y,    x-\-^y. 

The  advantage  of  this  latter  method  is,  that  the  sura  of 
the  series,  or  the  sum  or  difference  of  any  two  terms 
equally  distant  from  the  mean,  or  means,  will  contain  but 
one  unknown  quantity. 

1.  The  sum  of  three  numbers  in  arithmetical  progres- 
sion is  15,  and  the  sum  of  their  squares  is  83.  What 
are  the  numbers  ? 


224  ELEMENTARY   ALGEBRA. 

Let  X  =:  the  mean   term  and  y  =  the  common  difference  ; 
then  the  series  will  be  x  —  y,  x,  and  x  -\- 1/. 
By  the  conditions,  3x=rl5  (1) 

and  3x2  4- 2/ =  83  (2) 

Ans.  3,  5,  7. 

2.  The  sum  of  four  numbers  in  arithmetical  progres- 
sion is  44,  and  the  sum  of  the  cubes  of  the  two  means 
i.^  2926.  Ans.  6,  9,  13,  It. 

3.  Find  seven  numbers  in  arithmetical  progression  such 
that  the  sum  of  the  first  and  fifth  siiall  be  10,  and  the 
difference  of  the  squares  of  the  second  and  fourth  40. 

4.  There  are  four  numbers  in  arithmetical  progression ; 
the  product  of  the  first  and  third  is  20 ;  and  the  product 
of  the  second  and  fourth  84.     What  are  the  numbers  ? 

Ans.  2,  6,  10,  14. 

5.  The  sum  of  four  numbers  in  arithmetical  progression 
is  32 ;  and  their  product  3465.     What  are  the  numbers  ? 

Ans.  5,  7,  9,  11. 

6.  The  sum  of  the  squares  of  the  extremes  of  four  num- 
bers in  arithmetical  progression  is  461 ;  and  the  sum  of 
the  squares  of  the  means  425.     What  are  the  numbers? 

Ans.  10,  13,  16,  19. 

7.  A  certain  number  consists  of  three  figures  which 
are  in  arithmetical  progression ;  if  the  number  is  divided 
by  the  sum  of  its  figures,  the  quotient  will  be  15 ;  and 
if  396  is  added  to  the  .number,  the  order  of  the  figures 
will  be  inverted.     What  is  the  number  ?  Ans.  135. 

8.  Find  four  numbers  in  arithmetical  progression  such 
that  the  sum  of  the  squares  of  the  first  and  third  shall  be 
104,  and  of  the  second  and  fourth  232. 

9.  Find  four  numbers  in  arithmetical  progression  such 
that  the  sum  of  the  squares  of  the  first  and  second  shall 
be  29,  and  of  the  third  and  fourth  185. 


PROGRESSION.  225 

SECTION   XXIII. 

GEOMETRICAL    PROGRESSION. 

229.  A  Geometrical  Progression  is  a  series  in  which 
each  term,  except  the  first,  is  derived  by  multiplying  the 
preceding  term  by  a  constant  quantity  called  the  ratio. 

230.  If  the  first  term  is  positive,  when  the  ratio  is  a 
positive  integral  quantity,  the  series  is  called  an  ascending 
series,  and  when  the  ratio  is  a  positive  proper  fraction,  a 
descending  series  ;  but  if  the  first  term  is  negative,  the  se- 
ries is  ascending  when  the  ratio  is  a  positive  proper  frac- 
tion, and  descending  when  the  ratio  is  a  positive  integral 
quantity.     Thus, 

2,         6,       18,        54,  &c.)  ,. 

;>•  are  ascending  series  ; 
_54,  _i8,  _   6,  —    2,  &C.I  ^ 


16,         8,  &c.)  ,  ,. 

>-  are  descendine: 
—  32,  —64,  &c.  )  ^ 


64,       32, 

_ .'    ^      Y  are  aescendin^  series. 
8,  —  16, 


If  the  ratio  is  negative,  the  terms  of  the  progression  are 
alternately  positive  and  negative.  Thus,  if  the  ratio  is 
—  2  and  the  first  term  3,  the  series  will  be 

3,   —  6,   +  12,   —  24,   -f  48,  &c.  ; 
but  if  the  first  term  is  —  3, 

—  3,    +6,    —12,    4-24,    —48,  &c. 

The  positive  terms  of  these  two  series  constitute  an  as- 
cending progression  whose  ratio  is  the  square  of  the  given 
ratio  ;  and  the  negative  terms  a  descending  progression 
having  the  same   ratio. 

231.  In  Geometrical  Progression  there  are  five  elements, 
any  three  of  which  being  given,  the  other  two  can  be  found. 
These  elements  are  the  same  as  in  Arithmetical  Progres- 
sion, except  that  in  place  of  the  common  difference  we  have 
the  ratio. 

10*  *        o  ' 


226  ELEMENTARY   ALGEBRA. 

232.  Twent}'^  cases  may  arise  in  Geometrical  Progres- 
sion. In  discussing  these  cases  we  shall  preserve  the 
same  notation  as  in  Arithmetical  Progression,  except 
that  instead  of  d  ^  the  common  difference  we  shall  use 
r  =  the  ratio. 

CASE    I. 

233.  The  first  term,  ratio,  and  number  of  terms  given, 
to  find  the  last  term. 

In  this  Case  a,  r,  and  n  are  given,  and  I  required. 
The  successive  terms  of  the  series  are 

a,    ar,    ar^,    ai^^    a 7*,    &c. 
That  is,  each  term  is  the  product  of  the  first  term  and  that  power 
of  the  ratio  which  is  one  less  than  the  number  of  that  term  count- 
ing from  the  left ;  therefore  the  last  or  nth  term  in  the  series  is 


ar" 


or  l  =  ar"-^.     Hence, 

RULE. 

Multiply  the  first  term  by  that  power  of  the  ratio  whose 
index  is  one  less  than  the  number  of  terms. 

1.  Given  a  =  7,  r  =  3,  and  w  =  5,  to  find  /. 

l=ar-^-'^z=1  X  3^  =  667,  Ans. 

2.  Given  a  =  3,  r  =  2,  and  w  ::=  9,  to  find  /. 

Ans.  /=  768. 

3.  Given  a  =  64,  r  =  ^,  and  n  ^  10,  to  find  I. 

Ans.  /  =  i- 

4.  Given  a  =  —  7,  r  =.  —  4,  and  n  =  3,  to  find  /. 

Ans.  /  =  —  112. 

6.    Given  a  =  —  ^,  r  =  ^,  and  w  =  6,  to  find  /. 

Ans.  /=  —  j^^. 

6.  Given  a  =  5,  r  =  —  \,  and  w  =  10,  to  find  /. 

7.  Given  a  =  —  ^,  r  =  ^,  and  n  =  8,  to  find  /. 

8.  Given  a=:  —  10,  r  =  —  2,  and  n  =  6,  to  find  /. 


PROGRESSION.  227 

CASE   II. 

234.  The  extremes,  and  the  ratio  given,  to  find  the  sum 
of  the  series. 

In  this  Case  a,  I,  and  r  are  given,  and  S  is  required. 

Now  S  =  a -{- ar -\- ar" -\- ai^ -{- -^  I  (l) 

Multiplying  (1)  by  r,      r  S  =  ar-\- ar" -\- a?^ -}- ^IJ^lr  (2) 

Subtracting  (1)  from  (2),     r S  —  S  =  Ir  —  a 

Whence,  S= —.     Hence, 

r  —  1 

RULE. 
Multiply  the  last  tei^Tn  by  the  ratio,  from  the  product  sub- 
tract the  first  term,   and  divide  the  remainder  by  the  ratio 
less  one. 

1.  Given  az=2,  1=  20000,  and  r  =  10,  to  find  S. 

^^l_r-a^  20000  X  10  -  2  _  ^2222,  Ans. 
r  —  1  10  —  1 

2.  Given  a  =  1,  Z  =  45927,  and  r  =  3,  to  find  S. 

Ans.   ^r=  68887. 

3.  Given  a  =  —  5,  1=  —  405,  and  r  =  3,   to  find  *S'. 

4.  Given  a  =  —  t7T5)  ^  =  i>  and  r  =z  —  7,  to   find    S. 

Ans.   S=r,\% 

CASE    III. 

235.  The  first  term,  ratio,  and  number  of  terms  given, 
to  find  the  sum  of  the  series. 

In  this  Case  a,  r,  and  n  are  given,  and  5  required. 
The  last  term  can  be  found  by  Case  I.,  and  then  the  sum  of  the 
series  by  Case  11.     Or  better,  since 

lr'=  ar^ 

Substituting  this  value  of  /  r  in  the  formula  in  Case  II.  we  have 

r^  —  1 
S  = ~  X  a-     Hence, 


228  ELEMENTARY    ALGEBRA. 

RULE. 
From  the  ratio  raised  to  a  power  whose  index  is  equal  to 
the  number  of  terms  subtract   one,  divide  the  remainder   by 
the  ratio  less  one,  and  multiply  the  quotient  by  the  first  term. 

1.  Given  0  =  4,  r  =  7,  and  n  =  5,  to  find  S. 

S='^^^  Xa=  '^^  X  4  =  11204,  Ans. 

2.  Given  a  =  |,  r  =  5,  and  n  =  6,  to  find  S. 

Ans.   *S'=558. 

3.  Given  a  =  ^,  r  =  ^,  and  n  =  1,  to  find  S. 

Ans.   S=m. 

4.  Given  a  z=  —  6,  r  =  —  4,  and  w  =  4,  to  find  aS^. 

Ans.  .S'=255. 

5.  Given  a  =  —  |,  r  =  6,  and  n  =  5,  to  find  S. 

6.  Given  a  =  §,  r  =  —  3,  and  n  =  6,  to  find  S. 

7.  Given  a  =  —  },  r  =  2,  and  n  =  S,  to  find  S. 


In  a  geometrical  series  whose  ratio  is  a  proper  frac- 
tion the  greater  the  number  of  terms,  the  less,  numeri- 
cally, the  last  term.  If  the  number  of  terms  is  infinite,  the 
last  term  must  be  infinitesimal  ;  and  in  finding  the  sum 
of  such  a  series  the  last  term  may  be  considered  as  noth- 
ing. Therefore,  when  the  number  of  terms  is  infinite,  the 
formula 


S= becomes 


1  1  — r 

Hence,  to  find  the  sum  of  a  geometrical  series  whose  ra- 
tio is  a  proper  fraction  and  number  of  terms  infinite, 

RULE. 
Divide  the  first  term  by  one  minus  the  ratio. 


PROGRESSION.  229 


1     Find  the  sum  of  the   series  1,  ^,  |-,   &c.  to  infinit3^ 
'5  =  l^.=  l^i  =  2.  Ans. 

2.  Find  the  sum  of  the  series  -|,  |,   ^5,  &c.  to  infinity. 

Ans.  j%. 

3.  Find  the  sum  of  the  series  -,  ~^,  -,  &c.  to  infinity. 

.  1 

Ans. 


c  — 1 

4.  Find  the  sum  of  the  series  6,  4,  2|,  &c.  to  infinity. 

Ans.   18. 

5.  Find  the  value  of  the  decimal  .4444,  &c.  to  infinity. 

Note.  —  This  decimal  can  be  written  -^  -\-  j^  -\~  y^j^^,  &c. 

Ans.   |. 

6.  Find  the  value  of  .324324,   &c.  to  infinity. 

1.    Find  the  value  of  .32143214,   &c.  to  infinity. 

CASE    IV. 
237.    The  extremes  and  number  of  terms  given,  to  find 
the  ratio. 

In  this  Case  «,  /,  and  n  are  given,  and  r  is  required. 
From  Case  I.  Z  =  a;"-i 

Whence,  r=    »/-.     Hence, 

RULE. 

Divide  the  last  term  hj  the  first,  and  extract  that  root  of 
the  quotient  whose  index  is  one  less  than  the  number  of 
terms. 

1.    Given  a  ==  7,  /  =:  667,   and  n  =  5,  to  find  r. 


"-yi  4/567 


3,  Ans. 


2.    Given  a  ==  6|,  1=1,  and  n  =  6,  to  find  r. 

Ans.  r  =  ^. 


230  ELEMENTARY  ALGEBRA. 

3.  Given  a  =  —  ^,  /  =  31^,  and  n  =  4,  to  find  ••. 

Ans.  r  =  —  5. 

Note.  —  This  rule  enables  us  to  insert  any  number  of  geometri- 
cal means  between  two  numbers ;  for  the  number  of  terms  is  two 
greater  than  the  number  of  means.     Hence,  if  m  =  the  number  of 

m+l/J 

means,  m-|-2  =  n,  orw-|-l==n  —  1;    and  r  =   i/-«    Having 

found  the  ratio,  the  means  are  found  by  multiplying  the  first  term 
by  the  ratio,  by  its  square,  its  cube,  &c. 

4.  Find  three  geometrical  means  between  2  and  512. 

Ans.  8,  32,  128. 

5.  Find  four  geometrical  means  between  3  and  3072 

Ans.  12,  48,  192,  768. 

6.  Find  three  geometrical  means  between  1  and  y'^. 

Ans.  ^,  i,  i. 

Note.  —  When  m  =  1 ,  the  formula  becomes 

Multiplying  by  a,  ar  =  a  ^     -  =^1  al 

But  a  r  is  the  second  term  of  a  series  whose  first  term  is  a  and 
ratio  r ;  or  the  geometrical  mean  of  the  series  a,  ar,  a  r*.  Hence, 
the  geometrical  mean  between  two  quantities  is  the  square  root  of  their 
product. 

7.  Find  the  geometrical  mean  between  8  and  18. 

Ans.   12. 

8.  Find  the  geometrical  mean  between  ^  and  343. 

Ans.  7. 

9.  Find  the  geometrical  mean  between  ^  and  ji^^. 
10.  Find  the  geometrical  mean  between  —  ^  and  —  yiVr- 


PROGRESSION.  231 

238.  From  the  formulas  established  in  Arts.  233  and  234, 

l=ar^-^  (!)• 

S^^^-^  (2) 

r  —  1  ^  ^ 

can  be  derived  formulas  for  all  the  Cases  in  Geometrical 

Progression.  * 

From  (1)  we  can  obtain  the  value  of  any  one  of  the  four  terms,  /, 
a,  n,  or  r,  when  the  other  three  are  given;  frona  (2),  the  value  of  5, 
Z,  r,  or  a,  when  the  other  three  are  given.  Formulas  for  the  remaining 
twelve  Cases  which  may  arise  are  derived  by  combining  the  formulas 
(1)  and  (2)  so  as  to  eliminate  that  one  of  the  two  unknown  terms 
whose  value  is  not  sought. 

1.    Find  the  formula  for  the  value  of  *S',  when  /,  n,  and 
r  are  given. 
From   (1), 

Substituting  this  value  of  a  in  (2) 

or  (r_l)r»-i 

Note.  —  The  four  formulas  for  the  value  of  n  cannot  be  derived  or 
used  without  a  knowledge  of  logarithms ;  and  four  others,  when  n  ex- 
ceeds 2,  cannot  be  reduced  without  a  knowledge  of  equations  that  can- 
not be  reduced  by  any  rules  given  in  this  book. 

239.  To  find  any  one  of  the  five  elements  when  three 
others  are  given. 

RULE. 

Substitute  in  that  one  of  the  formulas  (1)  or  (2)  that  con- 
tains the  four  elements,  viz.  the  three  given  and  the  one  re- 
quired, the  given   values,  and  reduce  the  resulting  equation. 

If  neither  formula  contains  the  four  elements,  derive  a  for- 
mula that  will  contain  them,  then  substitute  and  reduce  the 
resulting  equation ;  or  substitute  the  given  values  before  deriv- 
ing the  formula,  then  eliminate  the  superfluous  element  *and 
reduce  the  resulting  equation. 


I 

V  — 

:  a 

ir          ' 

1 

r  —\ 
l(r^- 

0 

232  ELEMENTARY   ALGEBRA. 


1.    Given  r  —  3,  n  —  b,  and  S=  726,  to  find  I. 

l-af-'             (1)                     S=.^'-^ 

(2) 

/-81a               (3)                 726  =  '"-" 

(i) 

g^  =  a                     (5)                      a  =  3 ;  —  1452 

(6) 

8/- 1452=1                             (7) 

242  ;=r  1452  X  81             (8) 

/  =  486                         (9) 

Substituting  the  given  values  of  r,  n,  and  S  in  (1)  and  (2),  we 
obtain  (3)  and  (4)  ;  finding  the  value  of  a,  the  superfluous  element, 
from  (3)  and  (4),  and  putting  these  values  equal  to  each  other,  we 
form  (7),  an  equation  containing  but  one  unknown  quantity.  Re- 
ducing (7)  we  obtain  (9),  or  I  =  486. 

2.  Given  a  =  4,  r  =  5,  and  S=  15624,  to  find  I. 

Ans.   lz=z  12500. 

3.  Given  a  =  2,  w  =  5,  and  /=  512,  to  find  S. 

Ans.  >S'=682. 

4.  Find  the  formula  for  the  value  of  a,  when  r,  n,  and 

S  «^«  gi^e"-  Ans.  a  =  ^'~V- 

r»  —  1 

5.  A  gentleman  purchased  a  house,  agreeing  to  pay  one 
dollar  if  there  was  but  one  window,  two  dollars  if  there 
were  two  windows,  four  if  there  were  three,  and  so  on, 
doubling  the  price  for  every  window.  There  were  14 
windows.     How  much  must  he  pay?  Ans.  $8192. 

6.  A  man  found  that  a  grain  of  wheat  that  he  had  sown 
had  produced  10  grains.  Now  if  he  sows  the  10  grains 
the  next  year,  and  continues  each  year  to  sow  all  that  is 
produced,  and  it  increases  each  year  in  tenfold  ratio,  how 
many  grains  will  there  be  in  the  seventh  harvest,  and  how 
many  in  all?       ^^^      f  In  7th  harvest,  10000000  grains. 

tin  all,  1111 11 1 1  grains. 


PROGRESSION.  233 

PROBLEMS 
TO  WHICH  THE  FORMULAS   DO  NOT  DIRECTLY  APPLY. 

240.  In  solving  Problems  in  Geometrical  Progression, 
if  we  let  X  =  the  first  term  and  y  =  the  ratio,  the  series 
will  be 

X,  xy,  xy'^,  xy^,   &c. 

It  will  often  be  found  more  convenient  to  represent  the 
series  in  one  of  the  following  methods  :  — 

1st.   When  the  number  of  terms  is  odd, 


or  > 

',    ^y,  f) 


for  three  terms ; 

X' 

a?  if 

—,    x^,    xy,    7f,~  for  five  terms. 

y  x 


2d.   When  the  number  of  terms  is  even, 

a?  if 

-,    X,    y,    —  for  four  terms; 

y  ■^ 

3?    x"  y"  f  o     ' 

2?'    y'    ""'    ^'    x'   x^  ^'"^  *^^™®- 

Which  method  is  most  convenient  in  any  case  will  de- 
pend upon  the  conditions  that  are  given  in  the  problem. 

1.  There  are  three  numbers  in  geometrical  progression, 
the  greatest  of  which  exceeds  the  least  by  32  ;  and  the 
diflerence  of  the  squares  of  the  greatest  and  least  is  to 
the  sum  of  the  squares  of  the  three  as  80  :  91.  What  are 
the  numbers  ? 

SOLUTION. 

Let  X,  xy^  and  xy^  represent  the  series.     Then 

a;/  — x=32(l)    x*?/*  — x2:x2  +  x27/2-|-x2r/*==80:91  (2) 

y_l:l_|_y2^y4=80:91  (3) 

91y4_91=:80-(-80?/2-|-80?/*  (4) 

11  2^4—80  2/='=  171  (5) 

x=4     (7)  2/  =3  (6) 


234  ELEMENTARY   ALGEBRA. 

Dividing  the  first  couplet  of  (2)  by  x*,  we  obtain  (3) ;  from  (3) 
We  form  (4),  which  reduced  gives  (6),  or  y  =  3.  Substituting  the 
Value  of  y  in  (l),  we  obtain  (7),  or  x  ==  A. 

Ans.  4,  12,  36. 

2.  The  sura  of  three  numbers  in  geometrical  progres- 
sion is  39,  and  the  sum  of  their  squares  819.  What  are 
the  numbers  ? 

SOLUTION. 

Let  X,  \^xy,  and  y  represent  the  series.     Then 

x  +  ^Vy  +  y  =  S9  (1)         x^  +  xy  +  f  =  ^^^     (2) 

x-^JoJ+j^2i  (3) 

2  x^-\-2y  =  Q0  (4) 

x+     i/  =  SO  (5) 

2-v/'^=18  (6) 

xy  =  Sl  (1) 

Dividing  (2)  by  (1),  we  obtain  (3);  adding  (3)  to  (1),  we  ob- 
tain (4),  which  reduced  gives  (5)  ;  subtracting  (3)  from  (1),  we 
obtain  (6),  which  reduced  gives  (7).  Combining  (5)  and  (7)  as 
the  sum  and  product  are  combined  in  Example  1,  Art.  188,  we 
obtain  a:  =  27  and  y  =  3. 

Ans.  3,  9,  27. 

3.  Of  four  numbers  in  geometrical  progression  the  dif- 
ference between  the  fourth  and  second  is  60  ;  and  the  sum 
of  the  extremes  is  to  the  sum.  of  the  means  as  13  :  4. 
What  are  the  numbers  ? 

SOLUTION. 

Let  X,  xy,  xy^,  and  a;y*  represent  the  series.     Then 


xf  —  xy  —  m      (1) 

a;/  +  x:xy''  +  xy=13:4 

(2) 

y'-y+l:./=13:4 

(3) 

4^  — 4y  +  4  =  13y 

(4) 

64a:  — 4*  =  60     (7) 

4y»— 17y  =  — 4 

(5) 

x=  1        (8) 

y  =  4 

(6) 

PROGRESSION.  235 

Dividing  the  first  couplet  of  (2)  by  X7j  -\-  x,yve  obtain  (3)  ;  from 
(3)  we  form  (4),  which  reduced  gives  (6),  or  y  =  4.  Substituting 
this  value  of  y  in   (1)  and  reducing,  we  obtain  (8),  or  x=\. 

Ans.   1,  4,  16,  64. 

4.  Of  four  numbers  in  geometrical  progression  the  sum 
of  the  first  two  is  10  and  of  the  last  two  160.  What  are 
the  numbers  ?  Ans.  2,  8,  32,  128. 

6.  A  man  paid  a  debt  of  $310  at  three  payments.  The 
several  amounts  paid  formed  a  geometrical  series,  and  the 
last  payment  exceeded  the  first  b}^  $240.  What  were  the 
several  payments?  Ans.  $10,  $50,  $250. 

6.    In  the  series  x,  \/ xy,  and  y  what  is  the  ratio? 

Ans.  ^\ 

Y.    In  the  series  -,  x,  y,  and  -  what  is  the  ratio? 

8.  There  are  four  numbers  in  geometrical  progression 
whose  continued  product  is  64  ;  and  the  sum  of  the  series 
is  to  the  sum  of  the  means  as  5  :  2.  What  are  the  num- 
bers ?  Ans.  1,  2,  4,  8. 

9.  There  are  five  numbers  in  geometrical  progression  ; 
the  sum  of  the  first  four  is  156,  and  the  sum  of  the  last 
four  *r80.     What  are  the  numbers? 

10.  There  are  three  numbers  in  geometrical  progression 
whose  sum  is  126  ;  and  the  sum  of  the  extremes  is  to  the 
mean  as  1*7  :  4.     What  are  the  numbers? 

11.  The  sum  of  the  squares  of  three  numbers  in  geo- 
metrical progression  is  2275 ;  and  the  sum  of  the  ex- 
tremes is  35  more  than  the  mean.     What  are  the  numbers  ? 

12.  Of  four  numbers  in  geometrical  progression  the  sum 
of  the  first  and  third  is  52  ;  and  the  difference  of  the  means 
is  to  the  difference  of  the  extremes  as  5  :  31.  What  are  the 
numbers  ? 


23(3  p:lkmentary  algebra. 

SECTION   XXIY. 

MISCELLANEOUS    EXAMPLES. 

1.    From  6ac—  [>ab-{-c^  take  Sac—  {3a6  —  (c  — c^) 
+  '7c}.  Ans.    3ac  — 2a6  +  2c2-f6c. 


2.  Reduce  x^f  —  (—  xf  +  x^  —  ^)  xy  —  x^  (—   {/ 

—  y  {^1/  —  ^'0  1 )   to  its  simplest  form. 

Ans.  2x^f  +  x^. 

3.  Reduce  (a  —  b -\- cy  —  fa  (c  —  a  ^  b)  —   ^b  (a -\- b 

-\-  c)  —  c  (a  —  b  —  c) }  j   to  its  simplest  form. 

Ans.  2(a2  +  62  4-c2). 

4.  Reduce  (x  -{-  a)  a  -\-  i/  —   |  (^  -|_  ^)  (^x  +  b)  —  y 
(a:  +  a  —  1)  —  (x  -\-  y)  (b  —  a)}   to  its  simplest  form. 

Ans.  a^  —  i-. 
6.    Reduce  (a^  —  b"")  c  —  {a  —  b)  [a  {b-\- c) —b  {a  —  c)\ 
to  its  simplest  form.  Ans.  0. 

6.    Reduce  {a -{- b)  x— (b  —  c)  c  —  ^{b-^x)  b— {b  —  c) 
(5-|-c)}  —ax  to  its  simplest  form.       Ans.  2bx  —  be. 

1.    Multiply  a^  +  2aH  —  SaP  by  —  (—  3aH  -\-  aH^). 

8.  Multiply  a*  -f-  6  a2  _|_  9  by  a*  —  6  a^  +  9. 

9.  Multiply  a-\-  b  —  c  by  a  —  b -\-  c. 

10.  Divide  28a2  _  6a^  —  Ga^  —  4a^  —  96a  +  264  by 
3a2_4a  +  11. 

11.  Divide  1  —  18ar«4-  81a:*  by  1  -|-6x4-  9x\ 

12     Divide  9  0^+  1  —4a*  — 6  a  by  1  -f  2  a^  —  3  a. 
13.   Divide  9 x^  —  1  x^  f -}-  2/  by  S  x*  +  2x''i/  —  t/\ 


MISCELLANEOUS    EXAMPLES.  237 

14.  Divide  23  a  —  30  —  7  a^  +  6  a*  by  3  a  —  2  «=  —  6- 

15.  Find  the  prime  factors  of  a^  —  h^. 

16.  Find  the  prime  factors  of  4m^?^^  —  49m*w^^. 

17.  Find  the  prime  factors  of  .t^  —  ^xy-^-y"^. 

18.  Find  the  prime  factors  of  x^  —  ^^ 

19.  Find  the  greatest  common  divisor  of  5  ar* —  lOx^y 

-j-  15/  and  4.  x^  -\-  ^  x"^  y  -\-  ^  x  y'^  -\-  ^  y^ .      Ans.  x  -\-  y. 

20.  Find  the  greatest  common  divisor  of  8  «  i^  -f-  24  a  i* 
+  16  «  i  and  U'  +  U'  +  7  ^*  —  Ul  Ans.  ¥  -f  h. 

21.  Find  the  greatest  common  divisor  of  6  a:;^  -["  *^  ^V  — 
3y2  and  120^24- 22x^  +  6/. 

22.  Find  the  greatest  common  divisor  of  4  a;  -f-  4:  x'^  —  40 
and  ^x^y  —  48  y.  Ans.  x  —  2. 

23.  Reduce  7 —    ,^  ,  .,  ,    ^"\   ,    ..^^  to  its  lowest  terms. 


24.    Reduce    ,      ,, to   its   lowest    terms. 


(«- 

-6)  (a-2  4-2a6  + 

6^) 

a'- 

-3a2^,-|_3a6-  — 

^'  t 

a^  —  y' 

a^  — 2/* 

25.  Reduce  -r^, — ^—ri, — ^ — -1 — ^r  to  its  lowest  terms. 

26.  Find    the    least    common    denominator    and    reduce 

:; :; — i — ; — 5  —  :; — — ^  to  a  smglc  iraction. 

1  —  a        1-j-a        l-j-a'        1  —  a^  °  ^ 

Ans.  —  -— i — ^• 
1  -f-  « 

27.  Find    the    least    common    denominator    and    reduce 

1  _|_  ^2  1  _  „i2  .        ,        .         , . 

:; — —  .  —  - — i — 7i  to  a  Single  fraction. 
1  —  rrc         1  -[-  m-  ° 

28.  Find    the    least    common    denominator    and    reduce 

4a^-3a5~^-16a^-9a^6^  *°  ^  '"'^^^  ^'^^^'^"' 

29.  Reduce  to  one  fraction  with  the  least  possible  de- 

nominator  r r- ^^- ^ V-  -r-,  • 

0  bed  cd        '    hd 


238  ELEMENTARY   ALGEBRA. 

30.  Reduce  to  one  fraction  with  the  least  possible  de- 

.      .  a-\-b  b  4-c  ,  a-f-c 

nominator  7^ ^. r^  —  7 vt ix  +  n tt r;  • 

(b  —  c)  (c  —  a)        (a  —  c)  (a  —  b)    '   (/>  —  a)  {c  —  b) 

Ans.   yj -7 — ^ r-  =  0. 

(0  —  c)  (c  —  a)  (a  —  b) 

31.  Find    the    least    common    denominator    and    reduce 

32.  Reduce  to  one  fraction  with  the  least  possible  de- 

.      ,         I  -\-x  Ax  \  —  X  .  2a: -+62:' 

i^ominator  ^j^,  -  j— ^  -  jy^-^,.      Ans.  -p^:^.- 

33.  Reduce  a  —  c ^^ ^  to  its  simplest  form. 

I  +  -I1 

34.  Reduce ^^^  to  its  simplest  form. 

35.  Reduce    -— ?! 1- a;  and    — f— | 2y  each   to   a 

X  —  2y     ^  X  -\-2y  ^ 


single  fraction  and  find  their  product.         Ans.  j^-^ 


x» 


X 


36.  Subtract  -^—  from 

2^  — a;  a:  -hy 

X  X  —  ct 

37.  Subtract  Bx4--r  from  a: 


88.   Multiply  (-il)\y  ^/-g^- 

39.  Divide by  -; i,  and  multiply  the  result  by  a* 

a  —  X    '^  c?  —  ar  ^  ''  *' 

40.  Divide  '^^^-^^y.  by  -^^. 

41.  Divide  ^'  by  (-^^  +  -±-\ 


MISCELLANEOUS   EXAMPLES.  239 

42.  Divide  -7^7 — ;— t:v/  by  — r^- 

b  (a  -{-  by      "^    rt-  —  b- 

T^.    .1     a  A- X    ,     a  —  x^       a -\- x         a  —  x  , 

43.  Divide  — ^^- j —  by  — ' i — ,   and  give 

a  —  X    '     a-{-x     *'    a  —  x         a -\- x  ^ 

a"  +  x^ 
the  answer  in  its  lowest  terms.  Aus.  -—-^ 

d  Cl  X 

44.  Reduce ;— r  = 1  •     What  is  the  value  of 

a        a  -f-  0        a  —  0 

X,  if  a  =  —  2  and  6  =  3? 

45.  Reduce  x —  =  -  -j- 


2  7     '  2 

fj^c hx^ 

46.  Reduce  {a  -\-  x)  {h  —  x)  —  a  {b  —  c) ^ =  0. 

What  is  the  value  of  x,  if  a  =  2,  J  =  —  3,  and  c  ==  —  1. 

47.  Reduce     ~''    ^=- What  is  the  value  0/ 

b  a 

x,  if  a  =  2,  6  =r  —  1,  and  c  —  Zt 

\  A-  x 

48.  Reduce  a  —    ■— ^-  =  0. 

1  —  X 


49.  Find  the  value  of  x  in  the  equation  x  =  ^  "^ — ^— -^ 
in  its  simplest  form.  «  — *       «  +  * 

60.  A  man  spends  $2.  He  then  borrows  as  much  money 
as  he  has  left,  and  again  spends  $2.  Then  borrowing 
again  as  much  money  as  he  has  left,  he  again  spends  $2, 
and  then  has  nothing  left.  How  much  money  did  he  have 
at  first? 

61.  If  5  is  subtracted  from  a  certain  number,  two  thirds 
of  the  remainder  will  be  40.     What  is  the  number  ? 

52.  Having  a  certain  sum  of  money  in  my  pocket,  I 
lost  c  dollars,  and  then  spent  one  ath  part  of  what  re- 
mained and  had  left  one  6th  part  of  what  1  had  at  first.' 
What  was  the  original  sum  ?  What  does  the  answer  be- 
come if  a  =  3,  b  =z  9,  and  c  =  6  ? 


240  ELEMENTARY    ALGEBRA. 

53.  If  I  buy  a  certain  number  of  pounds  of  beef  at  SO. 25 
a  pound,  I  shall  have  $0.25  left;  but  if  I  buy  the  same 
number  of  pounds  of  lard  at  SO.  15  a  pound,  1  shall  have 
$1.25  left.     How  much  money  have  I? 

54.  Divide  84  into  three  parts  so  that  one  third  of  the 
first,  one  fourth  of  the  second,  and  one  fifth  of  the  third 
shall  be  equal. 

55.  In  a  certain  orchard  25  more  than  one  fourth  of 
the  trees  are  apple  trees,  2  less  than  one  fifth  are  pear 
trees,  and  the  rest,  one  sixth  of  the  whole,  are  peach 
trees.     How  many  trees  are  there  in  the  orchard  ? 

56.  A  merchant  spent  each  year  for  three  years  one 
third  of  the  stock  which  he  had  at  the  beginning  of  the 
year;  during  the  first  year  he  gained  $600,  the  second 
$500,  and  the  third  $400.  At  the  end  of  the  three 
years  he  had  but  two  thirds  of  his  original  stock.  What 
was  his  original  stock  ? 

57.  From  a  cask  of  wine  out  of  which  a  third  part  had 
leaked,  84  liters  were  drawn,  and  then  the  cask  was  half 
full.     What  is  the  capacity  of  the  cask  ? 

58.  A  gentleman  has  two  horses  and  a  chaise.  The 
chaise  is  worth  a  dollars  more  than  the  first  horse  and  h 
dollars  more  than  the  second.  Three  fifths  of  the  value  of 
the  first  horse  subtracted  from  the  value  of  the  chaise  is 
the  same  as  seven  thirds  of  the  value  of  the  second  horse 
subtracted  from  twice  the  value  of  the  chaise.  What  is 
the  value  of  the  chaise  and  of  each  horse  ?  What  are  the 
answers  if  a  =:  —  50  and  5  :=  50  ? 

59.  A  had  twice  as  much  money  as  B.  A  gained  $30 
and  B  lost  $40.  Then  A  gave  B  three  tenths  as  much  as 
B  had  left,  and  had  left  himself  20  per  cent  more  than  he 
had  at  first.     How  much  did  each  have  at  first  ? 


MISCELLANEOUS   EXAMPLES.  241 

60.  A  number  of  mea  had  done  one  third  of  a  piece  of 
work  in  9  days,  when  18  men  were  added  and  the  work 
completed  in  12  days.  What  was  the  original  number  of 
men  1 

61.  A  boatman  can  row  down  the  middle  of  a  river  14 
miles  in  2  hours  and  20  minutes ;  but  though  he  keeps 
near  the  shore  where  the  current  is  one  half  as  swift 
as  in  the  middle,  ft  takes  him  4  hours  and  40  minutes  to 
row  back.  What  is  the  velocity  of  the  water  in  the 
middle  of  the  river  ?  Ans.  2  miles  an  hour. 

62.  A  had  three  fifths  as  much  money  as  B.  A  paid 
away  $80  more  than  one  third  of  his,  and  B  $50  less 
than  four  ninths  of  his,  when  A  had  left  one  third  as  much 
as  B.     What  sum  had  each  at  first? 

63.  A  farmer  hired  a  man  and  his  son  for  20  days, 
agreeing  to  pay  the  man  $3.50  a  day  and  the  son  $1.25 
for  every  day  the  son  worked  ;  but  if  the  son  was  idle, 
the  farmer  was  to  receive  $0.50  a  day  for  the  son's  board. 
For  the  20  days'  labor  the  farmer  paid  $67.  How 
many  days  did  the  son  work  ? 

64.  I  purchased  a  square  piece  of  land  and  a  lot  of  three- 
inch  pickets  to  fence  it.  I  found  that  if  I  placed  the  pick- 
ets 3  inches  apart,  I  should  have  50  pickets  left ;  but  if  I 
placed  the  pickets  2^  inches  apart,  I  must  purchase  60 
more.  How  much  land  and  how  many  pickets  did  1  pur- 
chase ?  Ans.   18906^  square  feet  and  1150  pickets. 

65.  A  criminal  having  escaped  from  prison  travelled  10 
hours  before  his  escape  was  discovered.  He  was  then 
pursued  and  gained  upon  3  miles  an  hour.  When  his 
pursuers  had  been  on  the  way  8  hours,  they  met  an  ex- 
pressman going  at  the  same  rate  as  themselves,  who  had 
met  the  criminal  2  hours  and  24  minutes  before.  In  what 
time  from  the  commencement  of  the  pursuit  will  the  crimi- 
nal be  overtaken  ?  Ans.  20  hours. 

11  p 


242  ELEMENTARY    ALGEBRA. 

66.  In  February,  1868,  a  man  being  asked  the  time, 
answered  that  the  number  of  hours  before  the  close  of  the 
month  was  exactly  one  sixth  of  10  less  than  the  number 
that  had  passed  in  the  month.  What  was  the  exact 
time?  Ans.  February  25th,  10  o'clock,  p.  m. 

6t.  A  and  B  owned  adjoining  lots  of  land  whose  areas 
were  as  3  :  4.  A  sold  to  B  100  hectares  of  his,  and  after- 
ward purchased  of  B  two  fifths  of  B's  entire  lot ;  and  then 
the  original  ratio  of  their  quantities  of  land  had  been  re- 
versed.    How  much  land  did  each  own  at  first  ? 

Ans.  A,  300  ;  B,  400  hectares. 

68.  A  laborer  was  hired  for  70  days;  for  each  day  he 
wrought  he  was  to  receive  $2.25,  and  for  each  day  he  was 
idle  he  was  to  forfeit  $0.15.  At  the  end  of  the  time  he 
received  $118.50.     How  many  days  did  he  work? 

69.  A  sum  of  money  was  divided  equally  among  a  num- 
ber of  persons  by  giving  to  the  first  SI 00  and  one  sixth  of 
the  remainder,  then  to  the  second  $200  and  one  sixth  of 
the  remainder,  then  to  the  third  $300  and  one  sixth  of  the 
remainder;  and  so  on.  What  was  the  sum  divided  and 
what  the  number  of  persons  ? 

70.  A  besieged  garrison  had  a  quantity  of  bread  which 
would  last  9  days  if  each  man  rec^ved  two  hectograms  a 
day.  At  the  end  of  the  first  day  800  men  were  lost  in  a 
sally,  and  it  was  found  that  each  man  could  receive  2| 
hectograms  a  day  for  the  remainder  of  the  time.  What 
was  the  original  number  of  men  ? 

71.  Find  a  fraction  such  that  if  1  is  added  to  the  de- 
nominator its  value  will  be  J;  but  if  the  denominator  is 
divided  by  3  and  the  numerator  diminished  by  3,  its  value 
will  be  5. 

72.  If  7  years  are  added  to  A's  age,  he  will  be  twice  as 
old  as  B  ;  but  if  9  years  are  subtracted  from  B's  age,  he 
will  be  one  third  as  old  as  A.     What  is  the  age  of  each? 


miscp:llaneous  examples.  2  43 

T3.  A,  B,  and  C  compare  their  fortunes.  A  says  to 
B,  "Give  me  $700  of  your  money,  and  I  shall  have  twice 
as  much  as  you  retain."  B  says  to  C,  "Give  me  $1400, 
and    I    shall    have   three   times    as    much    as  you   retain." 

0  says  to  A,  "Give  me  $420,  and  I  shall  have  five  times 
as  much  as  you  retain."     How  much  has  each? 

H.  An  artillery  regiment  had  39  soldiers  to  every  5 
guns,  and  4  over,  and  the  whole  number  of  soldiers  and 
officers  was  six  times  the  number  of  guns  and  officers. 
But  after  a  battle  in  which  the  disabled  were  one  half  of 
those  left  fit  for  duty,  there  lacked  4  of  being  22  meri  to 
every  4  guns.  How  many  guns,  how  many  officers,  and 
how  many  soldiers  were  there  ? 

Ans.   120  guns,  44  officers,  940  soldiers. 

75.  A  car  containing  5  more  cows  than  oxen  was  started 
from  Springfield  to  Boston.  The  freight  for  4  oxen  was 
$2  more  than  the  freight  for  5  cows,  and  the  freight  for 
the  wdiole  would  have  amounted  to  $30  ;  but  at  the  end 
of  half  the  journey  2  more  oxen  and  3  more  cows  were 
taken  into  the  car,  in  consequence  of  which  the  freight  of 
the  whole  was  increased  in  the  proportion  of  6  to  5.  What 
was  the  original  number  of  cows  and  oxen,  and  what  was 
the  freight  for  each  ? 

9  cows  and  4  oxen. 

Freight  for  a  cow,  $2  ;  for  an  ox,  $3. 

76.  Two  sums  of  money  amounting  together  to  $1600 
were  put  at  interest,  the  less  sum  at  2  per  cent  more  than 
the  other.  ,If  the  interest  of  the  greater  sum  had  been  in- 
creased 1  per  cent,  and  the  less  diminished  1  per  cent,  the 
interest  of  the  whole  would  have  been  increased  one  fif- 
teenth ;  but  if  the  interest  of  the  greater  had  been  increased 

1  per  cent  while  the  interest  of  the  other  remained  the 
same,  the  interest  of  the  whole  would  have  been  increased 
one  tenth.     What  were  the  sums,  and  the  rates  of  interest? 

Ans.   $1200  at  7  per  cent ;   $400  at  9  per  cent. 


Ans.    -j 


244  ELEMENTARY   ALGEBRA. 

It.  A  and  B  can  perform  a  piece  of  work  together  in 
Hf  days.  They  work  together  10  days,  and  then  B  fin- 
ishes the  work  alone  in  16§  days.  How  long  would  it 
take  each  to  do  the  work  ? 

18.  The  Emancipation  Proclamation  of  President  Lincoln 
was  promulgated  on  the  let  day  of  January  in  a  year  rep- 
resented by  a  number  that  has  the  following  properties : 
the  second  (hundred's)  figure  is  equal  to  the  sum  of  the 
third  and  fourth  minus  the  first ;  or  to  twice  the  sum  of 
the  first  and  fourth  ;  the  third  is  a  third  part  of  the  sum 
of  the  four;  and  if  1818  is  added  to  the  number,  the  order 
of  the  figures  will  be  inverted.     What  was  the  year  ? 

Y9.  A  and  B  can  do  a  piece  of  work  in  a  days ;  A  and 
C  in  b  days  ;   B  and  C  in  c  daj'^s.     In  how  many  days  can 

each  do  it  ? 

• 

80.  A  can  do  a  piece  of  work  in  a  days,  B  in  J  days, 
and  C  in  c  days.  In  how  many  days  can  A  and  B  together 
do  it  ?  B  and  C  together  ?  A  and  C  together  ?  All  three 
together  ? 

81.  A  market-man  bought  some  eggs  for  $0.28  a  dozen, 
and  sold  some  of  them  at  3  for  8  cents  and  some  at  5  for  12 
cents,  receiving  for  the  whole  $6.24,  and  clearing  $0.64. 
How  many  did  he  sell  at  each  rate  ? 

82.  One  cask  contains  56  liters  of  wine  and  40  of  water, 
and  another  96  of  wine  and  16  of  water.  How  many  liters 
taken  from  each  cask  will  make  a  mixture  containing  52 
liters  of  wine  and  24  of  water? 

83.  A  and  B  are  travelling  on  roads  which  cross  each 
other.  When  B  is  at  the  point  of  crossing,  A  has  720  me- 
ters to  go  before  he  arrives  at  this  point,  and  in  4  minutes 
they  are  equally  distant  from  this  point ;  and  in  32  minutes 
more  they  are  again  equally  distant  from  it.  AVhat  is  the 
rate  of  each?      Ans.   A's,  100;  B's,  80  meters  a  minute. 


MISCELLANEOUS  EXAMPLES.  245 

84  Multiply  x"^  by  cc". 

85.  Multiply  x^  by  x'^, 

86.  Divide  y-^  by  y-^ . 

87.  Divide  a!'-''  by  a2+«. 

88.  Transfer  the  denominator  of  — f^s  to  the  numerator. 

89.  Free     _^  ^  _^  from  negative  exponents. 

X     y  z 

90.  Expand  {—2a^)\ 

91.  Expand  (a^h)'^. 

92.  Expand  (—3  x-2y"')4. 

93.  Expand  {x^  X  ^")'. 

94.  Expand  {x  —  \/^Y- 

95.  Expand  (a^  —  2  5)^ 

96.  Expand  {2x—yy. 

97.  Expand  (2  a!  — 3)^ 

98.  Expand  {^a  —  2b)\ 

99.  Find  five  terms  of  {x  —  y)-^. 

100.  Expand  [2  —  x  —  yf. 

101.  Expand  (3  —  «  —  6  +  c)^ 

102.  Find  ^~^. 

103.  Find 


104.  Find  \^  —  lQx\ 

105.  Find  the  square  root  of    ^—   /'2a +  2:^—4)  &/| 

4-a2  — 4a  -|-2ax  +  4  — 4x  +  a:2. 


106.    Reduce  </ 256  a^  &8  _  768  a^^^cUo  its  simplest  form 


246  ELEMENTARY   ALGEBRA. 

107.  Reduce  ly  -  and  tyl  to  equivalent  radicals  hav- 
ing a  common  index. 

108.  Add  V^.  V^,  and  x/J|. 

109.  From  a^IM  take  ^lO. 

110.  Multiply  i^fby  i^J. 

111.  Divide  —Sa/TO  by  ^/S". 

112.  Divide  >^'6  by  ^Q. 

113.  Find  the  cube  of  S  \/~2x. 

114.  Find  the  square  root  of  6/C^"3] 

115.  Multiply  6  +  \/^  by  3  —  \^J, 

116.  Expand  (x^  —  2  ^/^)^ 

m.    Expand  (^^_y-^y. 

118.  Expand  (^^-«-y. 

119.  The  area  of  a  rectangular  field  is  4  acres  and  35 
square  rods ;  and  the  sum  of  its  length  and  breadth  is 
equal  to  twice  their  difference.  What  are  the  length  and 
breadth  ? 

120.  Two  travellers,  A  and  B,  set  out  to  meet  each 
otlier.  They  started  at  the  same  time  and  travelled  on  the 
direct  road  toward  each  other.  On  meeting  it  appeared 
that  A  had  travelled  18  miles  more  than  B,  and  that  A 
could  have  travelled  B's  distance  in  9  days,  while  it  would 
have  taken  B  16  days  to  travel  A's  distance.  How  far 
did  each  travel?  Ans.   A,  72  miles;  B,  54  miles. 

121.  Find  three  quantities  such  that  the  product  of  the 
first  and  second  is  a  ;  of  the  second  and  third,  b ;  and  of 
the  first  and  third,  c. 


A  ,        /«c       ,        /ah       ,        /be 

Ans.  ±^/^,    ±y/,^.    ±y/- 


MISCKLLANKOUS   EXAxMPLES.  247 

122.  A  and  B  invest  in  stocks.  At  the  end  of  the 
year  A  sells  his  stocks  for  $108,  gaining  as  much  per 
cent  as  B  invested  ;  B  sold  his  for  $49  more  than  he 
paid,  gaining  one  fourth  as  much  per  cent  as  A.  What 
sum  did  each  invest?  Ans.   A,  $45;  B,  $140. 

123.  Reduce   18x'- —  33:r  —  40  =  0. 


124.    Reduce 


I  X  6 

336 


125.  Reduce  g-.y-(?-.)  =  f 

Ans.  y  =1  —  2.1  ±3^1-4:9,  or  5,  or  —  f. 

126.  Reduce  (x*  —  x"  ^  4)-^  +  x'  —  5704  +  x>. 
Ans.  x=r  ±3,  or  ±2 


V^,   or   ±^L±liz^, 


127.    Reduce  \/2a! -j-  1  +  2>v/^  = 


yjtx^ 


128.  Reduce  ^  —  2^/  — 3^  +  5  =  3^  —  2. 

129.  Reduce  — 


yj  x-\-1  y/ 


130.    Reduce  "'.,     ^f  +  ^/  ^  ^  _  2. 


131.  Reduce  (^+A^£^l«f  =  .  _  3. 

\:r  — v'r^—  16/ 

132.  Given    \    ^  ^^^^'^  ^^1  ,  to  find  x  and  y. 

133.  Given    |^  +  ^  ^  ^^  l  ,  to  find  a;  and  y. 

(^ZZ/  — 49) 

134.  Given    \  x  —  y  >-,  to  find  x  and  y. 

(      5  3:v— 75) 


248  ELEMENTARY   ALGEBRA. 


136.   Given    i  '        V  >• ,  to  find  x  and  v. 

136.  Given    f  5x^2^- 2x/=  875 1     ^^  ^^^  ^  ^^^ 

(  3a:y  =rl05i  ^ 

137.  Given    i  ^/  —  4ary  =:  96  >     ^^  ^^^  ^  ^^ 

(  x2  +  y2  =  25J  ^ 

138.  Given    •<  ^  r  ,  to  find  x  and  v. 

139.  Given    1'^''  + 2/' +  ^  +  y  =  12|    ^^  g^^  ^  ^^^  ^ 

Ans     1^===^'  or4(-3±V21). 
(^  =  2,  or  i 


140. 


l(-3:FV2i). 

Given    |^  +2^  =      ^U  ,  to  find  x  and  u. 
(x*4- v'  =  1921>  ^ 


141.  A  drover  sold  a  number  of  sheep  that  cost  him 
$297  for  $7  each,  gaining  $3  more  than  36  sheep  cost 
him.     How  many  sheep  did  he  sell  ? 

142.  A  merchant  sold  a  piece  of  cloth  for  $75,  gaining 
as  much  per  cent  as  the  piece  cost  him.  What  did  it 
cost  him  ? 

143.  A  drover  bought  12  oxen  and  20  cows  for  $920, 
buying  one  ox  more  for  $160  than  cows  for  $66.  What 
did  he  pay  a  head  for  each  ? 

144.  A  started  from  C  towards  D  and  travelled  4  miles 
an  hour.  After  A  had  been  on  the  road  6^  hours,  B 
started  from  D  towards  C,  and  travelled  every  hour  one 
fourteenth  of  the  whole  distance,  and  after  he  had  been 
on  the  road  as  many  hours  as  he  travelled  miles  an  hour, 
he  met  A.     What'was  the  distance  from  C  to  D  ? 


MISCELLANEOUS   EXAMPLES.  249 

145.  A  person  bought  a  number  of  horses  for  $1404. 
If  there  had  been  3  less,  each  would  have  cost  him  $39 
more.  What  was  the  number  of  horses  and  the  cost  of 
each? 

146.  Find  a  number  of  four  figures  which  increase 
from  left  to  right  by  a  common  difference  2,  while  the 
product  of  these  figures  is  384.  Ans.  2468. 

147.  A  rectangular  garden  24  rods  in  length  and  16  in 
breadth  is  surrounded  by  a  walk  of  uniform  breadth  which 
contains  3996  square  feet.  What  is  the  breadth  of  the 
walk?  Ans.   3  feet. 

148.  A  square  field  containing  144  ares  has  just  within 
its  borders  a  ditch  of  uniform  breadth  running  entirely 
round  the  field  and  covering  381.44  centares  of  the  area. 
What  is  the  breadth  of  the  ditch  ?  Ans.   0.8  meter. 

149.  A  and  B  hired  a  pasture  into  which  A  put  5  horses, 
and  B  as  many  as  cost  him  $5.50  a  week.  If  B  had  put 
in  4  more  horses,  he  ought  to  have  paid  $6  a  week.  What 
was  the  price  of  the  pasture  a  week?  Ans.   $8. 

150.  A  father  dying  left  $3294  to  be  divided  equally 
among  his  children.  Had  there  been  3  children  less,  each 
would  have  received  $  183  more.  How  many  children 
were  there  ? 

151.  A  merchant  bought  a  quantity  of  tea  for  $66.  If 
ho  had  invested  the  same  sum  in  coffee  at  a  price  $0.77 
less  a  pound,  he  would  have  received  140  pounds  more. 
How  many  pounds  of  tea  did  he  buy  ? 

152.  Find  two  quantities  such  that  their  sum,  product, 
and  the  sum  of  their  squares  shall  be  equal  to  one  an- 
other. Ans.  4  (3  ±  V"^^^)  and  i  (3  q=  \/^^^). 

153.  Find  two  numbers  such  that  their  product  shall 
be  6;  and  the  sum  of  their  squares  13. 

11* 


250  ELEMENTARY    ALGEBRA. 

154.  A  and  B  talking  of  their  ages  find  that  the  square 
of  A^8«age  plus  twice  the  product  of  the  ages  of  both  is 
3864  ;  and  four  times  this  product,  minus  the  square  of 
B's  age,  is  3575.     What  is  the  age  of  each? 

Ans.   A's,  42  ;  B's,  25. 

155.  Find  two  numbers  such  that  five  times  the  square 
of  the  less  minus  the  square  of  the  greater  shall  be  20  ; 
and  five  times  their  product  minus  twice  the  square  of  the 
greater  shall  be  25. 

156.  A  and  B  purchased  a  wood-lot  containing  600 
acres,  each  agreeing  to  pay  $17500.  Before  paying  for 
the  lot,  A  offered  to  pay  $20  an  acre  more  than  B,  if  B 
would  consent  to  a  division  and  give  A  his  choice  of  situ- 
ation. How  many  acres  should  each  receive,  and  at 
what  price  an  acre  ? 

Ans.   A,  250  acres  at  $  70  an  acre  ;  B,  350  at  $50. 

157.  A  merchant  bought  two  pieces  of  cloth  for  $175. 
For  the  first  piece  he  paid  as  many  dollars  a  yard  as  there 
were  yards  in  both  pieces  ;  for  the  second,  as  many  dol- 
lars a  yard  as  there  were  yards  in  the  first  more  than  in 
the  second  ;  and  the  first  piece  cost  six  times  as  much  as 
the  second.  What  was  the  number  of  yards  in  each 
piece?  Ans.  In  1st,  10  yards  ;  in  2d,  5. 

158.  Two  sums  of  money  amounting  to  $14300  were 
lent  at  such  a  rate  of  interest  that  the  income  from  each 
was  the  same.  But  if  the  first  part  had  been  at  the  same 
rate  as  the  second,  the  income  from  it  would  have  been 
$532.90;  and  if  the  second  part  had  been  at  the  same 
rate  as  the  first,  the  income  from  it  would  have  been 
$490.     What  was  the  rate  of  interest  of  each  ? 

Ans.  First,  7  per  cent ;  second,  7x(y  per  cent. 

159.  Divide  29  into  two  such  parts  that  their  product 
will  be  to  the  sum  of  their  squares  as   198  :  445. 


MISCELLANEOUS    EXAMPLES.  251 

160.  What  is  the  length  and  breadth  of  a  rectangular 
field  whose  perimeter  is  10  rods  greater  than  a  square 
field  whose  side  is  50  rods,  while  its  area  is  250  square 
rods  less  than  the  area  of  the  square  field  ? 

Ans.  Length,  75  rods  ;  breadth,  30. 

161.  A  rectangular  piece  of  laud  was  sold  for  $5  for 
every  rod  in  its  perimeter.  If  the  same  area  had  been 
in  the  form  of  a  square,  and  sold  in  the  same  way,  it 
would  have  brought  $90  less;  and  a  square  field  of  the 
same  perimeter  would  have  contained  272^  square  rods 
more.     What  were  the  length  and  breadth  of  the  field  ? 

Ans.   Length,  49  ;  breadth,  16  rods. 

162.  A  starts  from  Springfield  to  Boston  at  the  same^ 
time  that  B  starts  from  Boston  to  Springfield.  When 
they  met,  A  had  travelled  30  miles  more  than  B,  having 
gone  as  far  in  If  days  as  B  had  during  the  whole  time ; 
and  at  the  same  rate  as  before  B  would  reach  Springfield 
in  5|  days.     How  far  from  Boston  did  they  meet  ? 

Ans.  42  miles. 

163.  The  product  of  two  numbers  is  90  ;  and  the  dif- 
ference of  their  cubes  is  to  the  cube  of  their  difference  as 
13  :  3.     What  are  the  numbers? 

164.  A  and  B  start  together  from  the  same  place  and 
travel  in  the  same  direction.  A  travels  the  first  day  25 
kilometers,  the  second  22,  and  so  on,  travelling  each  day 
3  kilometers  less  than  on  the  preceding  day,  while  B 
travels  14^  kilometers  each  day.  In  what  time  will  the 
two  be  together  again  ?  Ans.  8  days. 

165.  A  starts  from  a  certain  point  and  travels  5  miles 
the  first  day,  7  the  second,  and  so  on,  travelling  each  day 
2  miles  more  than  on  the  preceding  day.  B  starts  from 
the  same  point  3  days  later  and  follows  A  at  the  rate  of 
20  miles  a  day.  If  they  keep  on  in  the  same  line,  when 
will  they  be  together  ?       Ans.  3  or  Y  days  after  8  starts. 


252  ELEMENTARY   ALGEBRA. 

166.  A  gentleman  offered  his  daughter  on  the  day  of 
her  marriage  $1000;  or  $1  on  that  day,  $2  on  the  next, 
$3  on  the  next,  and  so  on,  for  60  days.  The  lady  chose 
the  first  offer.  IIow  much  did  she  gain,  or  lose,  by  her 
choice  ? 

16*7.  The  arithmetical  mean  of  two  numbers  exceeds 
the  geometrical  mean  by  2  ;  and  their  product  divided  by 
their  sum  is  3^.     What  are  the   numbers  ? 

168.  A  father  divided  $130  among  his  four  children  in 
arithmetical  progression.  If  he  had  given  the  eldest  $25 
more  and  the  youngest  but  one  $5  less,  their  shares  would 
have  been  in  geometrical  progression.  What  was  the  share 
of  each  ? 

169.  The  sum  of  the  squares  plus  the  product  of  two 
numbers  is  133 ;  and  twice  the  arithmetical  mean  plus 
the  geometrical  mean  is  19.     What  are  the  numbers? 

110.  The  sum  of  three  numbers  in  geometrical  progres- 
sion is  111  ;  and  the  difference  of  the  second  and  third 
minus  the  difference  of  the  first  and  second  is  36.  What 
are  the  numbers  ? 

171.  There  are  four  numbers  in  geometrical  progression, 
and  the  sum  of  the  second  and  fourth  is  60  ;  and  the  sum 
of  the  extremes  is  to  the  sum  of  the  means  as  1  :  3.  What 
are  the  numbers  ? 


LOGARITHMS.  253 

SECTION   XXV. 

LOGARITHMS. 

241.  Logarithms  are  exponents  of  the  powers  of  some  num- 
ber which  is  taken  as  a  base.  In  the  tables  of  logarithms  in 
common  use  the  number  10  is  taken  as  the  base,  and  all 
numbers  are  considered  as  powers  of  10. 

By  Arts.  119,  120, 

10°=  1,     that  is,  the  logarithm  of     1  is  0 

10^=10,        "  "  10    "   1 

10^=100,      "  "  100    "  2 

10^=1000,    "  "  1000    "  3 

<fec.,  &c.,  &c. 

Therefore,  the  logarithm  of  any  number  between   1  and  10  is 

between  0  and  1,  that  is,  is  a  fraction ;   the  logarithm  of  any 

number  between  10  and  100  is  between  1  and  2,  that  is,  is  1 

plus  a  fraction  ;  and  the  logarithm  of  any  number  between  100 

and  1000  is  2  plus  a  fraction;  and  so  on.  ^ 

By  Art.  120, 

10**   =  1,        that  is,  the  logarithm  of  1.         is      0 
10-1  =  0.1,  "  "  0.1       "—1 

10-2  =  0.01,        ''  "  .    0.01     "  —2 

10-»  =  0.001,      "  «  0.001  "  —3 

&c.,  (fee,  &c. 

Therefore,  the  logarithm  of  any  number  between  1  and  0.1 
is  between  0  and  — 1,  that  is,  is  — 1  plus  a  fraction  ;  the 
logarithm  of  any  number  between  0.1  and  0.01  is  between — 1 
and  — 2,  that  is,  is  — 2  plus  a  fraction ;    and  so  on. 

The  logarithm  of  a  number,  therefore,  is  either  an  integer 
(which  may  be  0)  positive  or  negative,  or  an  integer  positive 
or  negative  and  a  fraction,,  which  is  always  positive. 


254  ELEMENTARY   ALGEBRA. 

The  reprcsentatioD  of  the  logarithms  of  all  numbers  less  than 
a  unit  by  a  negative  integer  and  a  positive  fraction  is  merely  a 
matter  of  convenience.  The  integral  part  of  a  logarithm  is 
called  the  characteristic,  and  the  decimal  part  the  mantissa. 
Thus,  the  characteristic  of  the  logarithm  3.1784  is  3,  and  the 
mantissa  .1784. 

242.  The  characteristic  of  the  logarithm  of  a  number  is 
not  given  in  the  tables,  but  can  be  supplied  by  the  following 

RULE. 
The  characteristic  of  the  logarithm  of  any  number  is  equal  to 
the  number  of  places  by  which  its  first  significaut  figure  on  the 
left  is  removed  from  units'  place,  the  characteristic  being  posi- 
tive when  this  figure  is  to  tJie  left  and  negative  when  it  is  to 
the  right  of  units'  place. 

Thus,  the  logarithm  of  59  is  1  plus  a  frjiction  ;  that  is,  the 
characteristic  of  the  logarithm  of  59  is  1.  The  logarithm  of 
5417.7  is  3  plus  a  fraction ;  that  is,  the  characteristic  of  the 
logarithm  of  5417.7  is  3.  The  logarithm  of  0.3  is  — 1  plus  a 
fraction ;  J:hat  is,  the  characteristic  of  the  logarithm  of  0.3  is 
— 1.  The  logarithm  of  0.00017  is  — 4  plus  a  fraction;  that 
is,  the  characteristic  of  the  logarithm  of  0.00017  is  — 4.. 

243.  Since  the  base  of  this  system  of  logarithms  is  10,  if 
any  number  is  multiplied  by  10,  its  logarithm  will  be  in- 
creased by  a  unit  (Art.  50) ;  if  divided  by  10,  diminished  by 
a  unit  (Art.  54). 

That  is,  the  log  of  5549  being       3.7442 

"     554.9  is       2.7442 


55.49 
5.549 
.5549 
.05549 
.005549 


1.7442 
0.7442 
T.7442 
2.7442 
3.7442 


LOGARITHMS.  255 

He7ice,  the  mantissa  of  the  logarithm  of  any  set  of  figures  is 
the  same,  wherever  the  decimal  point  may  he. 

As  only  the  characteristic  is  negative,  the  minus  sign  is 
written  over  the  characteristic. 


TABLE    OF    LOGARITHMS. 

244.  To  find  the  logarithm  of  a  number  of  two  figures. 

Disregarding  the   decimal  point,  find  the   given  number  in 

the   column   N    (pp.   268,   269),   and   directly  opposite,  in  the 

column  O,  is  the  mantissa  of  the  logarithm,  to  which  must  be 

prefixed  the  characteristic,  according  to  the  Rule  in  Art.  242. 

Thus,  the  log  of  85    is    L9294 

26    ''    L4150 

The  first  figure  of  the  mantissa,  remaining  the  same  for  sev- 
eral successive  numbers,  is  not  repeated,  but  left  to  be  supplied. 
Thus,  the  log  of  83  is  1.9191 

As,  according  to  Art.  243,  multiplying  a  number  by  10 
increases  its  logarithm  by  a  unit,  therefore,  to  find  the  loga- 
rithm of  any  number  containing  only  two  significant  figures 
with  one  or  more  ciphers  annexed,  we  use  the  same  rule  as 

above. 

Thus,  the  log  of      850   is   2.9294 

"    750000    "   5.8751 

The  principle  just  stated  is  applicable  also  in  the  cases  that 

follow. 

245.  To  find  the  logarithm  of  a  number  of  three  figures. 

Disregarding  the  decimal  point,  find  the  first  two  figures  hi 
the  column  N,  and  the  third  figure  at  the  top  of  one  of  the 
columns.  Opposite  the  first  two  figures,  and  in  the  column 
under  the  third  figure,  will  be  the  last  three  figures  of  the  deci- 
mal part  of  the   logarithm,  to  which  the   first   figure    in    the 


256  ELEMENTARY    ALCIEBRA. 

column  O  is  to  be  prefixed,  and  the  characteristic,  according 
to  the  Rule  in  Art.  242. 

Thus,  the  log  of  295    is   2.4698 
"      549    "    2.7396 

In  the  columns  1,  2,  3,  &c.,  a  small  cipher  (o)  or  figure  d)  is 
sometimes  placed  below  the  first  figure,  to  show  that  the  figure 
which  is  to  be  prefixed  from  the  column  O  has  changed  to  the 
next  larger  number,  and  is  to  be  found  in  the  horizontal  line 
directly  below. 

Thus,  the  log  of  7960   is   3.9009 
"     25900    "    4.4133 

246.  To  find  the  logarithm  of  a  number  of  more  than 
three  figures. 


On  the  right  half  of  pages  268  and  269  are  tables  of  Propor- 
tional Parts.  The  figures  in  any  column  of  these  tables  are 
as  many  tenths  of  the  average  difference  of  the  ten  logarithms 
in  the  same  horizontal  line  as  is  denoted  by  the  number  at 
the  top  of  the  column.  The  decimal  point  in  these  differences 
is  placed  as  though  the  mantissas  were  integral. 

1.  To  find  the  logarithm  of  a  number  of  four  figures,  find 
as  before  the  logarithm  of  the  first  three  figures;  to  this, 
from  the  table  of  Proportional  Parts,  add  the  number  stand- 
ing on  the  same  horizontal  line  and  directly  under  the  fourth 
figure  of  the  given  number. 

Thus,  to  find  the  log  of  5743. 

The  log  of  5740   is   3.7589 
In  "  Proportional  Parts,"  in  the  same  line,  under  3,  "  2.3 

Therefore,  the  log  of  5743    "    3.7591 

It  is  always  best  to  find  the  logarithm  of  the  nearest  tabu- 
lated number,  and  add  or  subtract,  as  the  case  may  be,  the 
correction  from  the  table  of  Proportional  Parts. 


LOGARITHMS.  257 


Thus,  to  find  the  log  of  6377. 

6377  =  6380  —  3 
The  log  of  6380   is   3.8048 

correction  for      3    "  2 


Therefore,  the  log  of  6377    "    3.8046 

Whenever  the  fractional  part  omitted  is  larger  than  half  the  unit  in 
the  next  place  to  the  left,  one  is  added  to  that  figure. 

2.  For  a  fifth  or  sixth  figure  the  correction  is  made  in  the 
same  manner,  only  the  point  must  be  moved  one  place  to  the 
left  for  the  fifth,  two  for  the  sixth,  figure. 

Thus,  to  find  the  log  of  3.6825. 

The  log  of  3.68  is  0.5658 

correction  for     2  "  2.4 

"  "      5  "  .59 


Therefore,  the  log  of  3.6825     '*    0.5661 

To  find  the  log  of  112.82. 

112.82  =  113  —  0.18 
The  log  of  113    is    2.0531 

correctionfor  .18  is  (3.8  +  3.02)    "  6.82 

Therefore,  the  log  of  112.82     "    2.0524 

The  logarithm  of  a  common  fraction  may  best  be  found  by 
reducing  the  fraction  to  a  decimal,  and  then  proceeding  as 
above. 

247.  To  find  the  number  corresponding  to  a  given  log- 
arithm. 

Find,  if  possible,  in  the  table  the  mantissa  of  the  given 
logarithm.  The  two  figures  opposite  in  the  column  N,  with 
the  number  at  the  head  of  the  column  in  w^hich  the  log- 
arithm is  found,  affixed,  and  the  decimal  point  so  placed  as 
to  make  the   number   of  integral    figures  correspond   to   the 


258  KLEMENTARY    ALGEBRA. 

characteristic  of  the  given  logarithm,  as   taught  in  Art.  242, 
will  be  the  number  required.     Thus, 

The  number  corresponding  to  log  5.5378  is  345000 

*•  1.8745  "  74.9 
If  the  mantissa  of  the  logarithm  cannot  be  exactly  found, 
take  the  number  corresponding  to  the  mantissa  nearest  the 
given  mantissa  ;  in  the  same  horizontal  line  in  the  table  of 
proportional  parts  find  the  figures  which  express  the  difference 
between  this  and  the  given  mantissa ;  at  the  top  of  the  page, 
in  the  same  vertical  column,  is  the  correction  that  belongs 
one  place  to  the  right  of  the  number  already  taken,  —  to  be 
added  if  the  given  mantissa  is  greater,  subtracted  if  less.    Thus, 

1.  To  find  the  number  corresponding  to 
log  2.7660 

next  less  log,    .2.7657,  and  number  corresponding,    583. 
difference,  3  correction,  0.4 

Number  required,  583.4 

2.  To  find  the  number  corresponding  to 
log  3.8052 

next  greater  log,  3.8055,  and  number  corresponding,  0.00639 
difference,  3  correction,  *         44 

Number  required,  0.0063856 

Tlie  nearest  number  in  the  table  of  Proportional  Parts  to  3  is  2.7  ; 
coiTesponding  to  this  at  the  top  is  4,  which  belongs  as  a  correction 
one  place  to  the  right  of  the  number  (0.00639)  already  taken,  but 
3  —  2.7  =  0.3  ;  this,  in  like  manner,  gives  a  still  further  correction  of 
4,  one  place  farther  still  to  the  light.  The  whole  correction,  there- 
fore, is  44,  to  be  deducted  as  shown  in  the  operation  al)ove. 

3.  Find  the  log  of  3764. 

4.  Find  the  log  of  2576000. 
6.    Find  the  log  of  7.546. 

6.    Find  the  log  of  0.0017. 


LOGARITHMS.  259 

7.  Find  the  log  of  ^. 

8.  Find  the  number  to  log  3.807873. 

9.  Find  the  number  to  log  1.820004. 

10.  Find  the  number  to  log  2.982197. 

11.  Find  the  number  to  log  2.910037. 

12.  Find  the  number  to  log  4.850054. 

248 •  The  great  utility  of  logarithms  in  arithmetical  opera- 
tions is  that  addition  takes  the  place  of  multiplication,  and 
subtraction  of  division,  multiplication  of  involution,  and  di- 
vision of  evolution.  That  is,  to  multiply  numbers,  we  add 
their  logarithms ;  to  divide,  we  subtract  the  logarithm  of  the 
divisor  from  that  of  the  dividend  ;  to  raise  a  number  to  any 
power,  we  multiply  its  logarithm  by  the  exponent  of  that 
power;  and  to  extract  the  root  of  any  number,  we  divide  its 
logarithm  by  the  number  expressing  the  root  to  be  found. 

This  is  the  same  as  multiplication  and  division  of  different 
powers  of  the  same  letter  by  each  other,  and  involving  and 
evolving  powers  of  a  single  letter  or  quantity ;  the  number 
10  takes  the  place  of  the  given  letter,  and  the  logarithms 
are  the  exponents  of  10. 


MULTIPLICATION    BY    LOGARITHMS. 

RULE. 

249t    Add  the  logarithms   of  the  factors,  and   the  sum  vrUl 
he  the  logarithm  of  the  product  (Art.  50). 

1.  Multiply  347.6  by  0.04752.  Ans.  16.518. 

2.  Find  the  product  of  0.568,  0.7496,  0.0846,  and  1.728. 

Ans.  0.06224. 
•    (It  must  be  carefully  borne  in  mind  that  the  mantissa  of 
the  logarithm  is  always  positive.) 


260  ELEMENTARY   ALGEBBA. 

3.    Multiply  0.00756  by  17.5. 

0.00756  log  3.8785 

17.5  ''     1.2430 


Product,    0.1323  "    1.1215 

4.  Multiply  0.0004756  by  1355. 

Although  negative  quantities  have  no  logarithms  (Art.  262), 
yet,  since  the  numerical  product  is  the  same  whether  the  fac- 
tors are  positive  or  negative,  we  can  use  logarithms  in  multi- 
plying when  one  or  more  of  the  factors  are  negative,  taking 
care  to  prefix  to  the  product  the  proper  sign  according  to 
Art.  48.  When  a  factor  is  negative,  to  the  logarithm  which 
is  used  n  is  appended. 

5.  Multiply  —0.7546  by  0.00545. 

—0.7546  log   1.8777  n 

0.00545  "     3.7364 


Product,  —0.004113  "     3.6141  n 

6.  Find  the  product  of  —0.017,  25,  and  —165.4. 

7.  Find  the  product  of  —14,  —7.643,  and  —0.004. 

Ans.  —.428. 

DIVISION    BY    LOGARITHMS. 

RULE. 

250.  From  the  logarithm  of  the  dividend  subtract  the  log- 
arithm of  the  divisor^  and  the  remainder  ivill  be  the  logarithm 
of  the  quotient  (Art.   54). 

1.    Divide  78.46  by  0.00147. 

78.46  log    1.8946 

0.00147  "     3.1673 


Quotient,  53374.  "     4.7273 


LOGARITHMS.  261 


2.    Divide  O.OOU  by  756. 

0.0014  log   3.1461 

756  "     2.8785 


Quotient,  0.000001852.        "     6.2676 

Negative  numbers   can  be  divided  in  the  same  manner  as 

positive,  taking    care    to    prefix   to  the    quotient  the   proper 
sign,  according  to  Art.  53. 

3.  Divide  0.7478  by  0.00456.  Ans.  164. 

4.  Divide  5000  by  0.00149. 

5.  Divide  0.00997  by  64.16.  Ans.  0.0001554. 

6.  Divide  —14.55  by  543.  Ans.  —0.0268. 

7.  Divide  —465  by  —19.45.  Ans.  23.9. 

251  •    Instead  of   subtracting  one   logarithm  from    another, 

it  is  sometimes  more  convenient  to  add  what  it  lacks  of  10, 

and   from    the    sum  reject  10.      The  result  is  evidently  the 

same.     For 

x  —  i/=zx -^{10  — 7/)  —  10 

The  remainder  found  by  subtracting  a  logarithm  from  10 
is  called  the  arithmetical  complement  of  the  logarithm,  or  the 
cologarithm.  The  cologarithra  is  easiest  found  by  beginning 
at  the  left  of  the  logarithm,  and  subtracting  each  figure  from 
9,  except  the  last  significant  figure,  which  must  be  subtracted 
from  10. 

By  this  method,  Ex.  1  will  appear  as  follows  : 
78.46  log         1.8946 

0.00147  colog    12.8327 

Quotient,  53374.  log        4.7273 


DIVISION   AND   MULTIPLICATION   BY   LOGARITHMS. 

252 1    In  working   examples    combining    multiplication   and 
division,  the  use  of  cologarithms  is  of  great  advantage. 


262  ELEMENTARY   ALGEBRA. 


RULE. 


Find  the  sum  of  the  logarithms  of  the  multipliers  and  the 
cologarithms  of  the  divisors ;  reject  as  many  tens  as  there  are 
cologarithms  (divisors) ;  the  residt  will  he  the  logarithm  of  the 
number  sought. 

^      r?-   A  ^x.         1  p  673  X  0.319  X  (—0.04) 

1.    Find  the  value  of  (,7.95)  ^  (-0.03478) 

673.  log  2.8280 

0.319  "  1.5038 

—0.04  "  2.6021  n 

^7.95  colog  9.0996  n 

—0.03478                   "  11.4587  n 


Ans.  —31.06  1.4922  n 

An  even  number  of  negative  quantities  gives  a  positive  re- 
sult, an  odd  number  a  negative  (Art.  48). 

2.  Find  the  value  of  ^I^fyigo.esT''''         ^"«-  ^'^'^' 

3.  Find  the  value  of  ^^^^^^^L^     Ans.  0.758. 


PROPORTION    BY    LOGARITHMS. 
RULE. 

253.    Add  the  cologarithm  of  the  first  term  to  the  logarithms 
of  the  second  and  third  termsj  and  from  the  sum  reject  *10. 

1.    Given  14  :  175  =  7486  :  x,  to  find  x. 

14  colog    8.8539 

175  log       2.2430 

7486  "        3.87425 


Ans.  93575  "        4.97115 

2.  Given  416  :  584  =  256  :  x,  to  find  x.         Ans.  359.3+ 

3.  Given  a;  :  179  =  49.68  :  489,  to  find  x,      Ans.  18.18+ 


LOGARITHMS.  263 

INVOLUTION    BY    LOGARITHMS. 

RULE. 

254 •     Multiply  the  logarithm  of  the  number  by  the  exponent 
of  the  power  required  (Art.    126). 

In    involution,  as  the   error  in  the  logarithm  is  multiplied 
by  the  index    of  the   power,  the    results  with   logarithms    of 
only  four  decimal  places   cannot   be   relied  on  for  more  than 
two  or  three  significant  figures. 
1.    Find  the  15th  power  of  1.17. 

1.17  log    0.0682 

15 


Ans.  1.0.54  "      1.0230 

Find  the  5th  power  of  0.00941. 

0.00941  log    3.9736 

5 


Ans.  0.0000000000738  "     11.8680 

3.  Find  the  4th  power  of  0.0176.     Ans.  0.00000009595. 

4.  Find  the  9th  power  of  1.179.  Ans.  4.49. 

Negative  numbers  are  involved  in  the  same  manner,  taking 
care  to  prefix  to  the  power  the  proper  sign,  according  to  Art.  125. 

5.  Find  the  3d  power  of —0.017.     Ans.  —0.000004913.  . 

6.  Find  the  6th  power  of  —14.  Ans.   7529536. 

In  the  last  two  examples  the  exact  answers  are  given, 
though  from  the  table  only  answers  approximating  to  these 
can  be  obtained. 

EVOLUTION    BY    LOGARITHMS. 

RULE. 
255 1     Divide  the  logarithm  of  the  number  by  the  exponent  of 
the  root  required  (Art.  143). 


264  ELEMENTARY   ALGEBRA. 

When  the  characteristic  is  negative,  and  not  divisible  by  the 
index  of  the  root,  we  increase  the  negative  characteristic  so  as 
to  make  it  divisible,  and  to  the  mantissa  prefix  an  equal  posi- 
tive number. 

1.  Find  the  5th  root  of  0.0173. 

0.0173  log  2.2380 

5)5  +  3.2380 
Ans.  0.4442  "  T.6476 

2.  Find  the  3d  root  of  80.07.  Ans.  4.31. 

3.  Find  the  8th  root  of  0.0764.  Ans.  ±0.725. 
Negative  numbers  are  evolved  in  the  same  manner,  taking 

care  to  prefix  to  the  root  the  proper  sign,  according  to  Art.  136. 

4.  Find  the  7th  root  of  —17.  Ans.  —1.499. 

5.  Find  the  5th  root  of  —0.00496.  Ans.  —0.346. 

MISCELLANEOUS     EXAMPLES. 
256.    Verify  the  following  expressions  : 

1748X_917^     7^  , 
654X513        *''*'-r 

2  —0.03479  X  2.3468'  X  i^«.843  _  j  359  i 
V^O.0678'*  X  (—4.63)  X  78.56 

3  ,V„,-^^-^^'''X  <^^"^£=-r= -0.2816+ 
'    y  ^673.21  X  50.3  X  V^).6845 

4.  ^/(^J2^^,^^^l^jr=  0.0001236+ 

V  V^O.OOOn  X  V0.00004782/ 

5.    f_^\_        =.0.000197+ 

331.9  (v/2.04  +  v/l.203)«  ' 

.     23.3X6.764x111^ 

'• 7.-....^'    =838.8+ 

X9.97 


^47.64  (I 


768\\ 
,853/ 


LOGARITHMS.  265 

SYSTEMS    OF    LOGARITHMS. 

257.  The  system  of  logarithms  which  has  10  for  its  base 
is  the  one  in  common  use.  As  in  this  system  the  mantissa 
of  the  logarithm  of  any  set  of  figures  is  the  same,  wherever 
the  decimal  point  may  be  (Art.  243),  which  (in  the  Arabic 
notation  of  numbers)  would  not  be  the  case  with  any  other 
base,  it  is  far  the  most  convenient  system.  The  number  of 
possible  systems,  however,  is  infinite. 

In  general,  if  a*  =  7i,  then  x  is  the  logarithm  of  n  to  the  base 
a\  and  n  is  the  number  (sometimes  called  the  antUogarithm) 
corresponding  to  the  logarithm  x,  in  a  system  whose  base  is  a. 

258.  The  logarithm,  of  1  is  0,  whatever  the  base  may  he. 
For  the  0  power  of  every  quantity  is  1,  or  a^  =  1  (Art.  120). 

259.  The  logarithm  of  the  base  itself  is  1. 

For  the  first  power  of  any  quantity  is  that  quantity 
itself,  or  a^  =  a  (Art.   119). 

260«     Neither  0  nor  1  can  be  the  base  of  a  system  of  logarithm's. 

For  all  the  powers  and  roots  of  0  are  0,  and  of  1  are  1. 

261*  The  logarithm  of  the  reciprocal  of  any  quantity  is  the 
negative  of  the  logarithm  of  the  quantity  itself. 

For  the  reciprocal  of  any  quantity  is  1  divided  by  that 
quantity  (Art.  27) ;  that  is,  is  the  logarithm  of  1  minus  the 
logarithm  of  the  quantity ;  or  0  minus  the  logarithm  of  the 
quantity  (Art.   250). 

262.  In  a  system  whose  base  is  greater  than  1,  the  log- 
arithm of  infinity  (  go  )  is  infinity ;  and  the  logarithm  of  0  is 
minus  infinity  ( — oo  ). 

For  a*  =  00  :    and  «"*=  -^  =  i  =  0. 

a^         CD 

Hence,  negative  quantities  cannot  have  logarithms. 

263  •  In  a  system  whose  base  is  between  1  and  0,  the  less  the 
number  the  greater  its  logarithm. 

For  the  greater  the  power  of  a  proper  fraction  the  less  its 


266 


ELEMENTARY   ALGEBRA. 


value.     With  such  a  base  the  logarithms  of  numbers  greater 
than  1  will  be  negative,  less  than  1  positive. 
Thus,  with  ^  as  the  base, 

the  log  of  ^  is      2  ;      of  ^  is       3 
"       9   "  — 2  ;      "   81   "   —3 

264  •  The  logarithms  of  numbers  which  form  a  geometrical 
series  form  an  arithmetical  series. 

For,  if  a  series  increased  or  decreased  by  a  constant  ratio, 
its  logarithms  would  increase  or  decrease  by  a  constant  dif- 
ference equal  to  the  logarithm  of  the  constant  ratio. 

For  an  example  see  Art.  243 ;  here  the  numbers  decrease 
by  the  constant  ratio  10,  and  the  logarithms  by  the  constant 
difference  1. 

265 1  From  the  principles  of  the  previous  articles  it  will 
be  easy  to  find  the  logarithms  of  the  perfect  roots  and  pow- 
ers of  any  number.     Thus, 

1.    In  a  system  whose  base  is  8, 

8^  =      2,  that  is,  the  log  of  2  =  0. 3 


8^=      4, 

tt 

"      4  =  0.6 

8^=     8, 
8^=    16, 
8^=   32, 

ii 

Cl 

It 

"      8=1. 
«    16  =  1.3 
«    32  =  1.6 

8'=   64, 

8^=128, 

&c.. 

u 
11 

&c., 

«    64  =  2. 
"128  =  2.3 
ifec. 

Then,  according  to  Art.  261, 

the  log  of  i  =  —0.3  =  r.6 
i  =  —0.6  =  L3 
-  i==~l.  =1. 
"  tV  =  — 1-3  =  2.6 
"  ^  =  —1.6  =  2.3 
"  ^  =  -2.  =2. 
"    Ti,  =  — 2.3  =  3.6 


LOGARITHMS.  267 

2.  In  a  system  whose  base  is  4,  what  is  the  logarithm  of 
41  of  161  of  641  of  21  of  81  of  1 1  of  ^1  of  ^1  of  ^1 
of  01 

3.  In  a  system  whose  base  is  9,  what  is  the  logarithm  of 
81  1   of  3  1   of  27  1   of  9  1   of  1 1    of  ^  1   of  ^^'^  of  01 

4.  In  a  system  whose  base  is  ^,  what  is  the  logarithm  of  2  1 
of  32  1  of  8  1  of  i  1  of  tV  ^ 

5.  If  the  logarithm  of  0.125  is  2.5,  what  is  the  basel 

a;-l=  0.125 


=  0.125  ^=  CqV=  8^  =  ^>  A^s- 


6.  If  the  logarithm  of  0.5  is  1.8,  what  is  the  base  1 

Ans.  32. 

7.  If  the  logarithm  of  0.3  is  0.3,  what  is  the  base  1 


EXPONENTIAL    EQUATIONS. 

266*  An  equation  having  the  unknown  quantity  as  an 
exponent,  or  an  eocponential  equation,  may  be  solved  by  means 
of  logarithms. 

For,  if  a'  =  ny  by  Art.   254, 

a;  X  log  a  =2  log  n 

log  n 
log  a 

1.    Solve  the  equation  125'=  25. 

xX  log  125  =  log  25 

_log   25__  1.3979 2     , 

*  ~  log  125^"  2:0969  ~"  3'         ^' 


2.  Solve  the  equation  2048'=  16. 

3.  Solve  the  equation  (0187)*=  ^7. 


268 


LOGARITHMS   OF  NUMBERS. 


1 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PROPORTIONAL   PARTS. 

1 

2 

3 

J^ 

5 

6 

7 

J_ 

9 

^ 

0000 

043 

086 

128 

170 

212253 

294 

334 

374 

4» 

8.3 

12.4 

16.6 

20.7 

24.8 

29.0 

331 

37-3 

11 

414 

453 

492 

531 

669 

607  645 

682 

719 

755 

38 

7.6 

"•3 

'5-1 

18.9 

22.7 

26.5 

30.2 

34-0 

12 

792 

828 

864 

899 

934 

969|o04 

„38 

oT' 

j06 

3-5 

7.0 

10.4 

13-9 

17.4 

20.9 

24.3 

27.8 

31.3 

13 

1139 

173 

206 

239 

271 

303  335 

367 

399 

430 

3.2 

6.4 

9-7 

12.9 

16. 1 

19-3 

22.5 

25-7 

29.0 

14 

461 

492 

523 

553 

584 

614644 

673 

703 

732 

30 

6.0 

9.0 

12.0 

15.0 

180 

21.0 

24.0 

27.0 

15 

1761 

790 

818 

847 

875 

903  931 

959 

987 

^ 

2.8 

76 

8.4 

II. 2 

14.0 

^8 

19.6 

22.4 

252 

16 

•2041 

068 

095 

122 

148 

175'201 

227 

263 

279 

2.6 

5-3 

7-9 

10.5 

132 

1S.8 

18.4 

21. 1 

237 

17 

304 

330 

356 

380 

405 

430  455 

480 

504 

529 

2.5 

S-o 

7-4 

9-9 

12.4 

»4-9 

»7.4 

19.9 

22.3 

18 

553 

577 

601 

625 

648 

672  695 

718 

742 

765 

2.3 

4-7 

7.0 

9-4 

II. 7 

14.1 

16.4 

18.8 

21.1 

19 

788 

810 

833 

856 

878 

900  923 

945 

967 

989 

2.2 

4-5 

67 

8.9 

II. I 

13.4 

15^ 

17^ 

20.0 

20 

3010 

032 

054 

076 

096 

118139 

160 

181 

•201 

2.1 

4.2 

6.4 

8.5 

10.6 

TT, 

14.8 

17.0 

19.1 

21 

22-2 

243 

263 

284 

304 

324  345 

365 

385 

404 

2  0 

4.0 

6.1 

8.1 

10. 1 

12.1 

14.1 

16.2 

18.2 

22 

424 

444 

464 

483 

502 

522  541 

660 

579 

598 

1.9 

3-9 

S.8 

77 

9-7 

11.6 

13-5 

154 

174 

23 

617 

636 

655 

674 

692 

711729 

747 

766 

784 

1.8 

3-7 

5-5 

74 

9.2 

11. 1 

12.9 

14.8 

16.6 

24 

802 

820 

838 

856 

874 

892  909 

927 

945 

962 

1.8 

35 

5-3 

7-1 

8.9 

10.6 

12.4 

14.2 

16.0 

26 

3979 

997 

7^ 

,31 

7» 

^te 

o99 

I'le 

l33 

I-7 

3-4 

5- 1 

6.8 

8.5 

10.2 

11.9 

136 

rri 

26 

4150 

166 

183 

200 

216 

232  249 

265 

•281 

298 

1.6 

3-3 

4-9 

6.6 

8.2 

9.8 

"•5 

I3-I 

.4.8 

27 

314 

330 

346 

362 

378 

393  409 

425 

440 

456 

1.6 

3-2 

4-7 

6-3 

7-9 

9-5 

II. I 

126 

14.2 

28 

472 

487 

502 

518 

633 

548  664 

579 

594 

609 

^■5 

3-0 

4.6 

6.1 

7.6 

9-' 

10.7 

12.2 

137 

29 

624 

639 

654 

669 

683 

698713 

728 

742 

757 

1.5 

_!:? 

4.4 

59 

7-4 

8.8 

lOJ 

11.8 

13-3 

30 

4771 

786 

800 

814 

829 

843  867 

871 

886 

900 

1.4 

2.8 

4-3 

5-7 

7-» 

"s^s 

10.0 

II. 4 

12.8 

31 

914 

928 

942 

955 

969 

983  997 

0" 

0^4 

„38 

1.4 

2.8 

4.» 

5-5 

6.9 

8.3 

9-7 

II.O 

12.4 

32 

6051 

065 

079 

092 

105 

119132 

146 

159 

172 

1.3 

2-7 

4.0 

5-3 

6.7 

8.0 

9-4 

10.7 

12.0 

33 

185 

198 

211 

224 

237 

250 -263 

276 

289 

302 

1-3 

2.6 

3-9 

5-2 

6.5 

7.8 

91 

10.4 

11.7 

34 

316 

328 

340 

353 

366 

3_78  3_91 

403 

416 

428 

1-3 

_2-5 

^ 

5^ 

_6j 

J± 

8.8 

10. 1 

"•3 

35" 

5441 

453 

465 

m 

490 

502514 

627 

539 

551 

1.2 

2-4 

3-7 

4.9 

6.1 

7-3 

86 

Ti 

II.O 

36 

563 

575 

587 

599 

611 

623  635 

647 

668 

670 

1.2 

2.4 

36 

4.8 

5.9 

7.1 

8.3 

9-5 

10.7 

87 

682 

694 

705 

717 

729 

740 1 752 

763 

776 

786 

1.2 

2.3 

3-5 

4.6 

5-8 

6.9 

8.1 

9-3 

10.4 

38 

798 

809 

321 

832 

843 

856  866 

877 

888 

899 

11 

2.3 

3-4 

4-5 

56 

6.8 

79 

9.0 

10.2 

89 

911 

922 

933 

944 

— 

955 

966  977 

988 

999 

0^0 

_!i' 

2.2 

3-3 

44 

5-S 

6.6 

7-7 

8.8 

99 

To 

6021 

031 

042 

053 

064 

075  085 

096 

107 

117 

I.I 

2.1 

3-2 

4-3 

5-4 

t; 

7-5 

8.6 

9-7 

41 

128 

138 

149 

160 

170 

180  191 

201 

212 

222 

I.O 

2.1 

3-1 

4-2 

5-2 

6.3 

7-3 

8.4 

9-4 

42 

232 

213 

253 

263 

274 

284  294 

304 

314 

325 

1.0 

2.0 

3» 

4  > 

51 

61 

7-2 

8.2 

9.2 

43 

335 

346 

355 

365 

375 

385  396 

405 

415 

426 

1.0 

2.0 

3-0 

4.0 

5-0 

6.0 

7.0 

8.0 

9.0 

44 

435 

444 

464 

464 

474 

484 '493 

503 

513 

5-2'2 

1.0 

2.0 

2.9 

3-9 

4-9 

5-9 

6.8 

-Zi! 

as 

Is 

6632 

542 

551 

iel 

671 

580  590 

599 

609 

618 

1.0 

1.9 

2.9 

.3.8 

4.8 

5  7 

6.7 

7.6 

Ti 

46 

628 

637 

646 

656 

665 

675 '684 

693 

702 

712 

0.9 

1.9 

2.8 

3-7 

47 

5-6 

6.5 

75 

8.4 

47 

721 

730 

739 

749 

758 

767  776 

785 

794 

803 

0.9 

1.8 

27 

3-7 

46 

5-5 

6.4 

7-3 

8.2 

48 

812 

821 

830 

839 

848 

857,866 

876 

884 

893 

0.9 

1.8 

27 

3-6 

4-5 

5-4 

6.3 

72 

8.1 

49 

902 

911 

920  928 

937 

946  955 

964 

972 

981 

0.9 

1.8 

2.6 

3-5    4-4 

5  3 

6.1 

70 

79 

jlo 

6990 

998 

o07  ol6 

7 

^y2 

^ 

^ 

o67 

0.9 

»-7 

2.6 

3.4JTI 

5-2 

6.0 

6.9 

7-7 

51 

7076 

084 

093  101 

no 

118  126 

135 

143 

152 

0.8 

1-7 

25 

3-4    4.2 

S» 

5-9 

6.7 

76 

62 

160 

168 

177  185 

193 

202  210 

218 

226 

235 

0.8 

17 

2.5 

3.3    4.. 

50 

5.8 

6.6 

7-4 

63 

243 

261 

269|q67 

276 

284  292 

300 

308 

316 

0.8 

1.6 

2.4 

3-2    41 

49 

S-7 

6.5 

7-3 

54 

324 

332 

340, 34S 

366 

36  4!  372 

^ 

388 

396 

0.8 

1.6 

2.4 

3-2    4.0 

4.8 

S-6 

6.4 

7.2 

LOGARITHMS   OF  NUMBERS. 


269 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PROPORTIONAL  PARTS. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

55 

7404 

412 

U9 

427 

m 

443 

47l 

469 

466 

474 

0.8 

1.6 

2.3 

31 

3-9 

4-7 

5-5 

6.3 

7.0 

56 

48-2 

490 

497 

505 

513  520 

528 

536 

643 

651 

0.8 

1-5 

2-3 

31 

3-8 

4.6 

5-4 

6.1 

6.9 

67 

559 

566 

574 

582 

689  597 

604 

612 

619 

627 

0.8 

1-5 

2-3 

30 

3-8 

4-5 

5-3 

6.0 

6.8 

58 

634 

642 

649 

657 

664 

672 

679 

686 

694 

701 

0.7 

1-5 

2.2 

30 

3-7 

4-5 

5-2 

S-9 

6.7 

59 

709 

716 

723 

731 

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270  ELEMENTARY  ALGEBRA. 


EXAMINATION   PAPERS   IN   ALGEBRA   FOR  ADMIS- 
SION TO  HARVARD  COLLEGE. 

June,  1878. 

1.  Two  workmen,  A  and  By  are  employed  on  a  certain  job  at 
different  wages.  When  the  job  is  finished,  A  receives  $  27,  and 
Bj  who  has  worked  three  days  less,  receives  $  18.75.  If  B  had 
worked  for  the  whole  time,  and  A  three  days  less  than  the  whole 
time,  they  would  have  been  entitled  to  equal  amounts.  Find 
the  number  of  days  each  has  worked,  and  the  pay  each  receives 
per  diem. 

2.  Find  the  value  of  x  from  the  proportion 

/lO  ^a^\  2         __     Iba^a''       96- « 
V  3^6V    •  ^  — V4^a2.6»  ''  ^5  * 

Express  the  answer  in  its  simplest  form,  free  from  negative  and 
fractional  exponents. 

3.  Simplify  the  expression 

ar^  -|-  /        a^  —  y^ 
x^  —  y"^       ar*  +  y« 

^  —  y  I  a^  +  y 

a:  +  y  "■    x  —  y 

4.  Write  out  the  first  five  terms  and  the  last  five  terms  of 

5.  Find  the  value  of  x  from  the  equations 

ax'\-hy=:.ly 
cy  -\'dz^=.my 
*ex-\-  fz  =zn. 


EXAMINATION   PAPERS.  271 

6.  Find  the  greatest  common  divisor  and  the  least  common 
multiple  oi6x^+7x  —  5aind2x^  —  x^  +  Sx—  4. 

7.  Solve  the  equation 

a;-)-13a4-36        a  —  2b_ 
5a  —  36  —  X        x-\-2b 

September,  1878. 

1.  Three  men,  A,  B,  C,  are  tried  on  a  piece  of  work.  It  is 
found  that  A  and  B  together  can  do  a  certain  amount  in  12  hours ; 
B  and  C  can  do  the  same  amount  in  8  hours  and  24  minutes ; 
and  C  and  A  can  do  the  same  amount  in  9  hours  and  20  min- 
utes. Find  the  time  which  each  man  would  require  to  do  the 
same  amount  singly. 

h  —  a        fa  —  2h        Zx{a  —  h)\ 

^Hh  ~~  \x-\-b  x'  —  b^J 

•  2.    Simplify 


b^-\-3b^x-{-3bx^  +  a^  _^        (x  +  bf 

a^  —  b"^  '    ar^  +  6ic-f-62 

3.  Write  out  the  first  five  terms  and  the  last  five  terms  of 

4.  Solve  the  equations,  ^{x  —  y)  =x  —  4,  a:y  =  2  ^  +  y  +  2. 

5.  Solve  the  equation,  x  -\ —  =  1  H jTa —  * 


a^-\-3b^ 
6.    Find  the  value  of  x  from  the  proportion, 

3  a 

262 


'•  ub^aK^b)  -^^  Vv^Ie^) 


Find  a  result  free  from  fractional  and  negative  exponents,  and 
in  the  most  reduced  form. 

7.    Find  the  greatest  common  divisor  and  the  least  common 
multiple  of  4:X*  —  x^—6x  —  9  and  Sx^'-^2x^—  Ix^—  6ic—  9. 


272  ELEMENTARY  ALGEBRA. 

June,  1879. 
1.    Solve  the  equation, 
2ax  —  46         bx  —  a  2abx 


bx  —  a  2ax  —  b'~  2abx^  —  (2a\r\-b^)x-\-ab* 

Eeduce  the  answers  to  their  simplest  forms. 

2.  Solve  the  equations, 

x     '       ^-tV 

4a:+32/=l. 
Stat&  clearly  what  values  of  x  and  y  go  together. 

3.  Pind  the  value  of  x  from  the  proportion 

36  Mh"^  ^      _  2(3qc)^  ,  2^a 
4    V     c»     •  ^~  ^/{y'c^)  '      be    \ 

4.  Simplify  the  fraction 

1  X  V 


X  —  y        a?'  —  y^        x^  -\-  y^ 

5.  Find  the  greatest  common  divisor  of 

2ar»_3a:+land2ar»  — x  — 1. 

6.  Put  the  following  question  into  equations  :  — 

A  and  B  walk  for  a  wager  on  a  course  of  one  mile  (5280  feet) 
in  length.  At  the  first  heat,  A  gives  B  a  start  of  45  seconds, 
and  beats  him  by  110  feet.  At  the  second  heat,  A  gives  B  a 
start  of  484  feet,  and  is  beaten  by  6  seconds.  Required,  the 
rates  at  which  A  and  B  walk. 


EXAMINATION   PAPERS.  273 

September,  1879. 

1.  Several  friends,  on  an  excursion,  spent  a  certain  sum  of 
money.  If  there  had  been  5  more  persons  in  the  party,  and 
each  person  had  spent  25  cents  more,  the  bill  would  have 
amounted  to  $  33.  If  there  had  been  2  less  in  the  party,  and 
each  person  had  spent  30  cents  less,  the  bill  would  have  amounted 
to  only  $  11.  Of  how  many  did  the  party  consist,  and  what  did 
each  spend  1     Find  all  possible  answers. 

2.  Solve  the  equations, 

2x-\-    4y  +  272  =  28, 
7x—    3y  — 152=3, 
9a;— lOy— 332=4. 

3.  Solve  the  equation, 

x-\-3b  3  6        __  a  +  36 

I    7 


8a^—Uab    '    4a^  — 96^        (2a+ 36)  (x  — 36) 
Reduce  the  answers  to  their  simplest  forms. 

4.    Calculate  the  sixth  term  of 


/      ^a      _      V2    Y 


Reduce  the  answer  to  its  simplest  form,  cancelling  all  common 
factors  of  numerator  and  denominator,  performing  the  numerical 
multiplications,  and  giving  a  result  which  has  only  one  radical 
sign  and  no  negative  or  fractional  exponents. 

5.  Simplify  the  fraction 

2a;  —  /        8a;«  — /' 
2^  +  /        8;«;8  +  i/« 

6.  Find  the  greatest  common  measure  and  the  least  common 
multiple  of 

4  ^6  _|_  14  rt  a;*  —  18  a^ic^  and  24  a^^  +  30  a«^  +  126  a*. 


274                                 ELEMENTAIIY  ALGEBRA. 
July,  1880. 
1.    Reduce  to  its  simplest  form 


x  + 


I-^-^ 


2x+l 


2.  Divide  60^"  +  ^"— 19:r^  +  2n_j_  20af«  +  »  —  7a:"»  —  4a:^-" 
by  3a;2«_5ar"+4. 

3.  Find  the  fourth  term  of  [^—'^ — — )    ,  reducing  it 

to  its  simplest  form. 

4.  Find  the  greatest  common  measure  and  the  least  common 
multiple  of  2a;5  — 11^2_9  .^^^  ^  x^ -\- U  x^ -\- ^\. 

5.  A  man  walks  2  hours  at  the  rate  of  4J  miles  per  hour. 
He  then  adopts  a  different  rate.  At  the  end  of  a  certain  time, 
he  finds  that  if  he  had  kept  on  at  the  rate  at  which  he  set 
out,  he  would  have  gone  three  miles  further  from  his  starting- 
point  ;  and  that  if  he  had  walked  three  hours  at  his  first  rate  and 
half  an  hour  at  his  second  rate,  he  would  have  reached  the  point 
he  has  actually  attained.  Find  the  whole  time  occupied  by  the 
walk  and  his  final  distance  from  the  starting-point. 


6.    Solve  the  equation 
•a  &(2a;-j-l) 


+ 


b(2x  —  l)        a{x*—l)        (2x— l)(x+l)    '    (2x— l)(a;— 1) 
Reduce  the  answers  to  their  simplest  forms. 

September,  1880. 
1.    Reduce  to  its  simplest  form  as  one  fraction 

/^^+  ?/  _i_  -r^  +  //^\  _^  (x  —  //  __  ^  —  ?/^\    ' 

V  —  ^'^ x^  +  fj  *  V+y      *'"  +  .vV* 


EXAMINATION   PAPERS.  275 

2.  Find  the  greatest  common  measure  and  the  least  common 
multiple  of 

3.  Find  the  sixth  term  of  (  —^ 6  <^6^  j   ,  reducing  the 

literal  part  of  the  term  to  its  simplest  form,  and  the  numerical 
part  into  its  prime  factors. 

4.  A  reservoir,  supplied  by  several  pipes,  can  be  filled  in  15 
hours,  every  pipe  discharging  the  same  fixed  number  of  hogs- 
heads per  hour.  If  there  were  5  more  pipes,  and  every  pipe 
discharged  per  hour  7  hogsheads  less,  the  reservoir  would  be 
filled  in  12  hours.  If  the  number  of  pipes  were  1  less,  and  every 
pipe  discharged  per  hour  8  hogsheads  more,  the  reservoir  would 
be  filled  in  14  hours.  Find  the  number  of  pipes  and  the  capacity 
of  the  reservoir. 

5.  Solve  the  equation 

2x-]-\  _  Zx-\-\  _  1  A  _  2\ 
b  a^  X  \b        aj ' 

Reduce  the  answers  to  their  simplest  forms. 

June,  1881. 

1.  What  are  the  factors  of  x^-\-  i/^1     Reduce  to  its  simplest 

^ 7/2  y 

form  the  product  of    o    ■      3  and  — r r 

P        a? 

2.  Solve  the  equation  - — : ; —  =  1  H 1 — * 

1  -f-a  +  ^  ^    a    '   X 

3.  Find  the  square  root  of 

1  _|_  y'Te^  +  10;;c"*  +  12  V^  +  9^"*. 


276  ELEMENTARY  ALGEBRA. 

4.  What  is  meant  by  the  expression  a?  1 

5.  Solve  the  equation  ^x  —  8  —  ^x  —  3  =  ^/x, 

6.  A  man  rows  down  a  stream,  of  which  the  current  runs  3^ 
miles  an  hour,  for  1§  hours.  He  then  rows  up  stream  for  6^ 
hours,  and  finds  himself  two  miles  short  of  his  original  starting- 
place.     Find  his  rate  and  the  distance  he  rowed  down  stream. 

7.  Find  the  4th  and  the  14th  term  of  (2  a  —  hf. 


EXAMINATION   PAPERS.  277 


EXAMINATIONS    FOR   ADMISSION    TO    YALE 
COLLEGE. 


June,  1878.. 
L  (a)  Reduce  — — — 5 5 to  its  lowest  terms. 

(b)  Multiply  a-^b^  by  -^ ;  and  divide  a'^b^  by  ~. 

2.    Solve  the  equations  : 

7a;  —  6  x  —  5  x 


{«) 


35  6a;— 101        5 


...    7*4-9        /         2a:— 1\ 

3.  (a)  Solve  the  equation  —  —  --|-7|=8. 

2        3 

(6)  It  is  required  to  find  three  numbers  such  that  the  product 
of  the  first  and  second  may  be  15,  the  product  of  the  first  and 
third  21,  and  the  sum  of  the  squares  of  the  second  and  third  74. 

4.  Find  the  sum  of  n  terms  of  the  series  1,  2,  3,  4,  5,  6,  &c. 

5.  By  the  binomial  theorem  expand  to  five  terms  (a^  —  b^)-k 

September,  1879. 

1.  Add  together  ^^s^^._^^.y  ^^e  J_  ^,y  2a*(a*  +  xi 

2.  (a)  Multiply  together  ^y' 3,  ^^3,  and  ^^3. 
(b)    Divide  9  m^  (a  —  6)i  by  3  m  (a  —  b)i. 


278  ELEMENTARY  ALGEBRA. 


3.    Solve  the  equations 

,    .      ^  X  —  4:  ,  5x  +  U  1 

(6)    --|_-  =  a;  _4._==^,.      __(__  — c. 

X       y  X        z  y        z 


4.    Solve  the  equation 

15        72  — 6a; 


2. 


«  2a:2 

5.  Find  the  sum  of  20  terms  of  the  series  1,  4,  10,  20,  35,  &c. 

6.  By  the  binomial  theorem  expand  to  4  terms 

(a)   (l-A)-i;  (6)    (a^-ar")*. 

July,  1880. 

1.  (a)  Divide  __j  +  -^  by  -— ^  -  ^-p^,  and  re- 
duce  the  quotient  to  its  simplest  form. 

if)  Find  the  greatest  common  divisor  of 

x^—  6x2  — 8a:  — 3  ^nd  4x»—  12a:  — 8. 

2.  (a)  Find  the  sum  of  6  V^47^  2  'V^2rt,  and  \^^\ 
{h)  Reduce  to  its  simplest  form  the  product 

(a:-l  -  A/^)(a:-l  +  V^)  (a:- 2  +  \/^)(ar- 2  -  a/^). 

3.  Solve  the  equations 

(a)   \{2x  _  10)  -  Ti^(3a:  -  40)  =  15  —  ^(57  -  x) ; 

(6)    ._1  +  -A_  =  0; 


W 


ar'  — 1 


=  ^+i. 


EXAMINATION   PAPERS.  279 

4.  Four  immbers  are  in  arithmetical  progression  ;  the  product 
of  the  first  and  third  is  27,  and  the  product  of  the  second  and 
fourth  is  72.     What  are  the  numbers  1 

5.  By  the  binomial  theorem  expand  to  4  terms, 

(a)    (1-6)-^  (b)    (x'-y')K 

September,  1880. 
1.  (a)  Eequired,  in  its  simplest  forhi,  the  quotient  of 
a*  —  X*  _    a^  X  -{-  x^ 


€?  —  2ax  -\-  x^    '     ci?  —  dt? 
(6)  Find  the  greatest  common  divisor  of 

6a;2—  17x+  12  and  12a;2  _  4a;  —  21. 

2.  Find  the  sum  of 

^l6,    ^^,    —  v'dsB,    ^^192,    —  7'V^9. 

3.  Solve  the  equations 

2  a- 1  :r  4-  2 

(a)    5x_^^^  +  l==3:.  +  ^  +  7; 

(6)        3;z:2_j_  |o^_57_0; 

/  s  a;2         ;r     ,     1         1 

(c)  — =  — 

^^  3         10    "^6        5 

4.  Find  three  geometrical  means  between  2  and  162. 

5.  By  the  binomial  theorem  expand  to  4  terms. 


280  ELEMENTARY  ALGEBRA. 

July,  1881. 

1.  Free  from  negative  exponents  (Aa~^b^ar*)~*. 

^ 2ar 15 

2.  Keduce  to  lowest  terms 


a:2-|-10a;  +  21 

3.  Factorw^— 2ra''4-?i;   ar*  —  1 ;   a^  — nV;   a;«  + /. 

2 

4.  Make  denominator  rational  of  — =:^ —> 

>v/5  —  a/2 

5.  Multiply  v^  —  2  +  ^^^Z  by  V^  +  2  —  V^Ts. 


6.    Solve —  , 


5  __  3  a:  4-1  _  1 

X  c^  4 

7.  Solve  01?  —  xy=  153 ;         x  -\-  y  =^\, 

8.  By  the  Binomial  Theorem  expand  to  four  terms 


9.    Sum  the  infinite  series  1  -|-  -  -|-  -  -|-  &c. 


EXAMINATION   PAPERS.  281 


EXAMINATIONS    FOR    ADMISSION    TO    AMHERST 
COLLEGE. 

June,  1878. 

,     -r^.   .,  a*  —  m*  -      a^ -\- am 

1.  Divide  ^ j :,  by  '■ 

^2 I 

2.  Reduce  —, r  to  its  lowest  terms. 

ab  —  0 

I  _l_  ^ 

3.  Given  b  —  ,  =  0  :  to  find  x. 

1  —  X 

4.  2x-{-37/=2'd;  5x'-'2i/=10;  findxandi/. 

5.  Find  the  cube  root  of  a^  VI? 


6.  Divide  4  V^Iac  by  2  VSa. 

7.  A  father's  age  is  twice  that  of  his  son  ;  but  10  years  ago 
it  was  three  times  as  great.     What  is  the  age  of  each  1 

8.  If  1  be  added  to  the  numerator  of  a  fraction,  its  value  is 
^  ;  and  if  1  be  added  to  the  denominator,  its  value  is  J.  What 
is  the  fraction  1 

September,  1878. 

1.  Resolve  1  —  36  y'^  into  two  factors. 

2.  Find  the  least  common  multiple  of  9a^,  12a^x^,  and 
24:ax^i/. 

-.    ,  -       ,    a       ,  —  X  -4-  d 

3.  Find  the  sum  of  a;  -|-  r-  and 


b  in 


4.    Divide  —. by  — ' — - 


282  ELEMENTARY  ALGEBRA. 


I X 

5.    Given  8a  =  -. — j — ,  to  find  x. 

I  "f-  X 


6.  Reduce  7  v9a^  —  27 a^b  to  its  simplest  form. 

7.  Find  the  square  root  of  4a*  —  12 a^  -\-  5 a^  -\-  6 a  -\-  I, 

8.  What  numbers  are  those  whose  difference  is  20  and  the 
quotient  of  the  greater  divided  by  the  less  is  3 1 

June,  1879. 

1.  Reduce  3  a  —  (2  a  —  [a  -|-  2])  to  its  simplest  form. 

2.  Find  the  greatest  common  divisor  oi2a^  —  7ar*-|-5j;  —  6 
and3a.^  — 7;r2  — 7a;4-3. 


1  —  X         1  -\-  X 
3.    Reduce  — —  to  its  simplest  form. 

+ 


l^x ' l+x 

4.  Resolve  a^  —  Ifi  into  four  factors. 

11^ 3 

5.  Given  7x =  3a:  +  7;  find  x. 

6.  A  crew  can  row  20  miles  in  2  hours  down  stream,  and  12 
*  miles  in  3  hours  against  the  stream.    Required  the  rate  per  hour 

of  the  current,  and  the  rate  per  hour  of  the  crew  in  still  water. 

7.  Extract  the  square  root  of  dx*  —  l2a^-\-l6x^— Sx -{- i. 

8.  Express  2  x^i/^  —  3  ar"  V* — x~*ir^  with  positive  exponents. 


9.  Divide  af-^hjx    *     and  reduce  the  quotient  to  its  sim- 
plest form. 

10.  Multiply  a-\-b  ^{—  1)  by  a  —  6  ^(—  1). 


EXAMINATION  PAPERS.  283 

September,  1879. 

1.  Interpret  a^  j  aP ;  a~^ ;  ai 

2.  Multiply  a  —  6  by  c  —  c?,  and  deduce  the  rule  that  "  like 
signs  give  -j-  ^^^  unlike  give  — ." 

3.  Separate  into  prime  factors  3m*x  —  3n*x. 

4.  Explain  the  reason  of  the  following  equations  : 

a^'a''  =  «"*  +  ";   a""  -i-  a""  =  a*"-" ;   (a"*)"  =  a"«. 

^     2x  —  9       x—3,x        25  —  3x     ^    ^ 

5.  ^^ ^+-=-^— ;findx. 

6.  A  left  a  certain  town  at  the  rate  of  a  miles  an  hour,  and 
in  n  hours  was  followed  by  B  at  the  rate  of  b  miles  an  hour. 
In  how  many  hours  did  B  overtake  A  1 

7.  The  sum  of  two  numbers  is  s  and  the  difference  is  d.  What 
are  the  numbers  1  Show  from  the  result  that  if  from  the  greater 
of  two  numbers  you  subtract  one  half  the  sum,  the  remainder 
will  be  one  half  the  difference. 

8.  Find  the  square  root  of  x*  —  4a^y  -|-  6 x^y^  —  4:xy^ -\-  ^. 

9.  Simplify  7  ^(a2»62"»«c^). 
10.    Subtract  3  ^a«  from  6  ^a\ 

June,  1880. 


1.  Reduce  (a  +  6  —  c)  \x-\-y  —  (a  -|-  J  -|-  c)  {x -\-  y)\  to 
its  simplest  form. 

2.  Resolve  c^  —  W  into  its  prime  factors. 

3.  Divide  -3-^  by 


66Mc£2    '  h^ddPe 


284  ELEMENTARY  ALGEBRA. 

,     ^x  -\-  2a       X  —  5a        ^        ,,    , 

4.   ^ =  5  a ;  nnd  x, 

A  o 

5.  What  numlDer  multiplied  by  m  gives  a  product  a  less  than 
n  timea  the  number  ] 

6.  5^-[-3y=19,    7a;  — 2y  =  8;  finda;andy. 

7.  Find  the  square  root  oi  a  -\-  la^x^  -\-  x. 

8.  Find  the  square  root  of  81a^a;~^ylr"i. 

9.  Free    _^  ^  from  negative  exponents,  and  reduce  the 

c      ~~~  ct 

result  to  its  simplest  form. 

10.  Multiply  a  4-  6  ^T^l  hj  a  —  b  V^^l. 

September,  1880. 

1.  Factor  81  a»  —  l. 

2.  Find  the  greatest  common  divisor  of 

a'  —  a^x  -\-  Sax^  —  3x^,  and  a^  —  5aa:  -|-  ix^. 

3.  Reduce  a  -\-  b to  a  fractional  form. 

'  a  —  b 

find  X  and  y. 

5.    Find  the  square  root  of  9a;*  —  12a:8  +  I6x^  —  8a;  +  4. 


6.  Find  the  sum  of  ^/3aH  and  VS¥b. 

7.  Multiply  5ai  by  3  ah. 

8.  Find  the  cube  of  a  —  bxh. 


EXAMINATION   PAPERS.  285 


9.    Find  the  cube  root  of  (x  -\-  y)  \^x  -f-  y. 
10.    Simplify  «-+A^  +  ^-^A^p. 


June,  1881. 


1.  Eemove  the  parentheses  and  reduce  to  its  simplest  form 
a  —  \a  —  [a  —  (a  —  x)]j. 

2.  Find  the  greatest  common  divisor  of  8a^-j-2a;  —  3  and 
120^34-10^^  —  4. 

3.  Resolve  ar^^  —  y^^  into  its  simplest  factors. 

Ou  X  X 

4.  Reduce  -5 -I 7—-  — •  to  its  simplest  form. 

x^  —  l'ic  +  1        1  —  ^ 

6.    Find  the  square  root  of 

^a^—  12^/+  \Qa?y^  —  UxUf  -\-  iy^  -\-\&xf. 

6.  y  —  a=z2{x  —  h) ;  y  —  h  z=z2{x  —  a) ;  find  x  and  y. 

7.  Multiply  Y4a  by  'v/G^. 

8.  A  wine-merchant  has  two  kinds  of  wine  which  cost  90 
cents  and  36  cents  a  quart,  respectively.  How  much  of  each 
must  he  take  to  make  a  mixture  of  100  quarts  worth  50  cents  a 
quart  1 

9.  Divide  xl -^  x\  -^  Q,  hy  xh  ^  2. 

— 3 —  )         *^  ^*^  simplest  form. 

11.  \i  a  '.h  =.  c  :  d,  prove  a  -\~h  -.a  —  h  =.  c  -\'  d  :  c  —  d. 

12.  3;r2  — 4a;=7;  find^ir. 


286  ELEMENTARY  ALGEBRA. 

13.  Insert  three  geometrical  means  between  13  and  208. 

14.  Demonstrate  the  two  fundamental  formulae  of  Arithmeti- 
cal Progression. 

15.  What  is  the  Binomial  Theorem?    Give  the  first  four 
terms  of  {a  -\-  xy^. 


EXAMINATION   PAPERS.  287 


ENTRANCE    EXAMINATIONS    TO    DARTMOUTH 
COLLEGE. 

1878. 

1.  Define  term,  factor,  coefficient,  exponent,  power,  root,  equa- 
tion. What  is  the  degree  of  a  term  ]  When  is  a  polynomial 
homogeneous  1 

2.  Write  the  following  without  using  the  radical  sign : 

3.  Write  the  following  without  using  negative  exponents  : 


V 


4.   Multiply  a  —  6  V —  1  by  «  +  ^  V —  1«    Also  a  —  6  V —  1 
by  a  4"  ^  V —  1- 


5.  Raise  a  —  6  v —  1  to  the  3d  power.     Simplify  the  radi- 
cal (a^  —  2  a^ft  +  a  i^)*. 

6.  Solve  ^^^^  —  ^^-^  =  6.     Also4-,  +  <^^°  +  c=:0. 

a  -\-  X         a  —  X  X   ^ 

..     X  —  1        X  —  2        X  -\-\        ..      ah  —  (a  —  x^        1 

Also  — —  =  — ^- — .     Also ^ '-  =  -. 

2  3  6  ah  +  {a  —  x)^       a 

1880. 

1.  Define  Algebra,  factor,  coefficient,  exponent,  fraction,  equa- 
tion. 

2.  Write  the  following  without  using  the  radical  sign  : 


288  ELEMENTARY   ALGEBRA. 

3.  Write  the  following  without  using  negative  exponents  : 

4.  Write,  in  the  simplest  form,  the  values  of 

5.  Find  the  product  of  ^/ab  X  —  "^J  X  (—  a*a:i)  X  2 a6 ; 
also  of  (2  +  V^^)  (2  —  V^^). 

6.  Solve 

X           ,          X  ,      ar— 2       .301        ^^,  ^       ar  — 2 

b  = ;  also  — - —  +  — —  =  .001  a;  +  .6  — 


a  +  1  a  - 1 '  5       '     .5  '  .05 

1881. 

1.  Define  term,   factor,    coefficient,    exponent,   power,   root, 
equation. 

2.  What  is  the  degree  of  a  term  1 
When  is  a  polynomial  homogeneous  1 

3.  Write  the  following  without  using  the  radical  sign : 

\^;         v^?;         Va'  +  62  — 2a6. 

4.  Write  the  following  without  using  negative  exponents  : 

x~^:         ax-^ :         ^ 

b-' 

5.  Multiply  a  —  c  V —  1  hy  a  ~\-  b  V —  1- 

6.  Raise  6  —  a  V —  1  to  the  3d  power. 
Simplify  the  radical  \/(^  —  2<^b  -^  cb\ 


EXAMINATION   PAPERS.  289 


7.  Solve r-^ =  a. 

a-\-  y  a  —  y 

X  —l        X  —  2        ^4-1 

8.  Solve  — — —  =- 


G 


a 

X 


9.    Solve  ;^  —  6;r«  +  c  =  0. 


290  ELEMENTARY   ALGEBRA. 


EXAMINATIONS    FOR    ADMISSION    TO    BROWN 
UNIVERSITY. 

1878. 
c 


—  1 


c  —  \ 
5.    Reduce to  a  simple  fraction. 


c  +  l 

6.  Divide  a  into  two  parts,  such  that  m  times  one  shall  be  n 
times  the  other. 

7.  If  4  be  subtracted  from  both  terms  of  a  fraction,  the  value 
will  be  ^ ;  and  if  5  be  added  to  both  terms,  the  value  will  be  ^. 
What  is  the  fraction  1 


8.  Given  a/o:.-  —  9  +  Va;  +  1 1  =  10,  to  find  x. 

^     ^.        rt      ,    3^  —  6        _  3a:  —  3        -    , 

9.  (jriven  zx  -\ =z  bx — ,  to  nnd  x. 

'2  X  —  6 


1879. 
^      ...       1  x-\-\  .  ^«+^+ 1 

2.  Multiply  {a  +  bf  by  (a  +  6)?,  and 

Divide    {a  +  6)  Va^TT'l    by   {a  —  i)  a/«*  +  2  a  +  1, 
giving  answers  in  simplest  forms. 

3.  Given,    "'<"  +  ->    ,- r^^,t-h,  =  ''^i^^ 

{a  —  b){x  —  a)       (a  —  b)  {x  —  6)  or  —  or 

to  find  X. 


4 


4.    Given  ^'1  -\-  x  -\-  ^/x  =.  —  ,  to  find  x. 

a/2  +ar 


EXAMINATION    PAPEKS.  291 

5.  Divide  the  number  s  into  two  such  parts,  that  if  m^  be 
divided  by  the  second,  and  this  quotient  multiplied  by  the  first, 
the  product  is  the  same  as  if  7i^  be  divided  by  the  tirst  and  the 
quotient  multiplied  by  the  second. 

1880. 

1.  Find  the  H.  C.  D.  of 

6^:3  _  g^^2  _^  2fx  and  l'2x^  —  I5x,>/  +  3/. 

2.  Given  ax-^bf/=zc  and  mx  =zni/ -\-dj  to  find  x  and  y. 

3.  Extract  the  cube  root  of 

_  99:z;3  _  9^.5  _j_  -^8  _^  (34  _  144^  _^  i^q^  _|_  39^4^ 

4.  Given  — — ^ -=-  =^  . ■—,  to  Imd  x. 

5.  Two  pipes,  A  and  B,  will  fill  a  cistern  in  70  minutes,  A 
and  C  in  84  minutes,  and  B  and  C  in  140  minutes.  How  long 
will  it  take  each  to  fill  it  alone  % 

6.  Given  V^  +  x  -\-  \/x  =  —  ,  to  find  x. 

V5  +  x 

7.  A  gentleman  bought  two  pieces  of  silk  which  together 
measured  36  yards.  Each  cost  as  many  shillings  per  yard  as 
there  were  yards  in  the  piece,  and  the  cost  of  the  pieces  were 
to  each  other  as  4  to  1.  Required  the  number  of  yards  in  each 
piece. 


8.  Given  x -\-\/5x -{- 10  =  S,  to  find  x. 

9.  Given  sr^  —  ^0  =  50,  to  find  x. 

10.    Given  a^  -\-  xi/  =  15  and  x^  —  ?/^  =  2,  to  find  x  and  y. 


14  DAY  USE 

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